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Copyright N°_ 

COPYRIGHT DEPOSIT; 



THE THERMODYNAMICS OF 
HEAT-ENGINES 



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THE THERMODYNAMICS 
OF HEAT-ENGINES 



BY s 

SIDNEY A. REEVE 

PROFESSOR OF STEAM-ENGINEERING AT THE WORCESTER 
POLYTECHNIC INSTITUTE 






Neto If orfe 
THE MACMILLAN COMPANY 

LONDON : MACMILLAN & CO., LTD. 
1903 

All rights reserved 






THt LIBRARY OF 
CONGRESS, 

Two Copies Received 

IAN 12 1903 

ft Copyrignt Entry 
CLASS CU XXe, No 

s o \> i % 

copy s,_. : . 



Copyright, 1903, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped January, 1903. 



x 



3 




Wortoootr $ress 

J. S. Cushing & Co — Berwick & Smith 
Norwood Mass. U.S.A. 



Eo JHg MiU 



to whose devotion and aid 

(although she doesn't know entropy from carbonic acid) 

the existence of this book is due 

it is dedicated 



CONTENTS 

PART I 

THEORY 
CHAPTER I 

PAGE 

The General Principles of Energetics i 

Definitions. The Forms of Energy. Their Interchangeability. 
The Law underlying Such Changes. The Constancy of Energy. 
The First Law of Energetics. The First Law of Thermodynamics. 
The Dimensions of Energy. Classification. The Direction of 
Energy-transformation. The Second Law of Energetics. The 
Second Law of Thermodynamics. 

CHAPTER II 

The Cycle -. ..*... 37 

Definitions. The Conditions Essential to the Existence of Cycles. 
The Rectangular Cycle in Mechanical Energies. The Rectangular 
Cycle in Thermodynamic Energies. The Force-distance and Entropy- 
temperature Diagrams. Isotherms and Adiabatics. The Isomor- 
phic Curve. Efficiencies. The Reversible Cycle. The Refrigerat- 
ing Cycle. 

CHAPTER III 

The Thermal Properties of Matter 66 

The Molecular Hypothesis. Molecular Energies. Solids, Liquids, 
and Gases. Specific Heat. Latent Heats. Temperature-heat. 
Disgregation-energy and External Work. The Absolute Zero. 
The Portrayal of Thermal Processes by the Entropy-temperature 
Diagram. The Formation of Steam under Constant Pressure. 

vii 



viii CONTENTS 



CHAPTER IV 

PAGE 

The Steam-engine Cycle .90 

The Triangular Cycle of the Boiler-explosion. The Quadrilateral 
Cycle of the Perfect Steam-engine, or the Rankine Cycle. Modifi- 
cations due to Wet Steam and to Superheat. Effect of Exhaust 
into a Vacuum'. The Cycle of the Non-expansive Steam-engine, or 
of the Direct-acting Steam-pump. The Standard Steam-engine 
Cycle, with Partial Expansion. The Carnot Cycle with Steam as a 
Working-substance. 

CHAPTER V 

The Laws of the Permanent Gases 113 

Boyle's Law. Charles's Law. The Two combined. The Natural 
Significance of the Constant. Expansion-curves in the Pressure- 
volume Diagram. Their Equations. The Work done by them. 
Alterations in Temperature. The Compression and Expansion 
of Air. 

CHAPTER VI 

The Gas-engine Cycles 131 

The Lenoir Cycle. The Beau-de-Rochas or Otto Cycle. The 
Joule Cycle. The Natural Relations between the Three. Efficien- 
cies and Capacities. 

CHAPTER VII 

The Hot-air Engines 152 

The Stirling Cycle. The Ericsson Cycle. The Carnot Cycle 
with Air as a Working-substance. Efficiencies, Capacities, and 
Possibilities. 

CHAPTER VIII 

Comparative Summary of Heat-engine Possibilities . . .164 



CHAPTER IX 

The Refrigerating Machines . . . . . . . .167 

The Cycle of the Air-refrigerating Machine, or the Reversed 

Joule Cycle. The Cycle of the Ammonia Compression Machine. 
The Kelvin Warming-machine. 



CONTENTS ix 

PART II 

APPLICATION TO PRACTICE 
CHAPTER I 

PAGE 

The Simple Steam-engine . . .185 

Projected Indicator-cards. Elementary Calculations. Expansion- 
ratios and Clearance. 

Thermal Modifications of the Theoretic Cycle in the Actual 
Cylinder. Condensation and Reevaporation. Factors affecting it. 
Speed. Size. Ratio of Expansion. Steam-jacketing. Superheat. 
The Observation and Analysis by the Entropy-temperature Method 
of the Cycle followed by the Actual Engine. Efficiencies. 

Resultant Types of Steam-engine. The Single-valve, Fixed- 
eccentric, Throttle-governed Engine. The Single-valve Shaft- 
governed Engine. The Four-valve Type. Comparison of Results 
and of Proper Fields of Service. 

CHAPTER II 

The Compound Steam-engine 240 

Mechanical Considerations. Thermodynamic Considerations. 
Cylinder-ratios. Woolf and Receiver Types. Methods of Governing. 

CHAPTER III 

The Otto Gas-engine 271 

Modifications of the Theoretic Cycle in the Actual Cylinder. The 
Effect of the Water-jacket. The Effect of Delayed Combustion. 
The Effect of the Cylinder-walls on the Expansion-curve. Simi- 
larity of the Thermal Phenomena in Gas-engines and Steam-engines. 
Scavenging. 

The Observation and Analysis by the Entropy-temperature 
Method of the Cycle followed by the Actual Gas-engine. 

APPENDIX 307 



NOTATION 



C = the density-constant for a permanent gas, or the number of foot-pounds 

of external work done during the heating of one pound of the gas 

one degree Fahrenheit under constant pressure. For air, C= 53.18. 

C = the specific density-constant involved in a case concerning a mass of 

gas other than one pound. 
D = the disgregation-energy of dry saturated steam. 
E = quantity of energy. 
F= efficiency. 

H ' = the total heat of dry saturated steam above that of melted ice. 
/= internal energy of a gas or vapor. 
J — the mechanical equivalent of heat = 778. 
K = the ratio of the specific heats of a gas under constant pressure and 

constant volume respectively. 
L = the latent heat of vaporization of dry saturated steam. 
/ = the isomorphic heat of water above that of melted ice. 
M = mass. 
N= entropy. 

P= pressure, in pounds per square inch unless otherwise stated. 
Q = quantity of heat. 

Qi = the heat supplied at the upper temperature-level during a cycle. 
(?2 = the heat rejected at the lower temperature-level of a cycle. 
Q w — the heat available for doing work = Q\ — Q%. 
q w = the heat actually converted into work during a cycle. 
R = volume-ratio of gaseous expansion. 
.S = specific heat. 

T = absolute temperature Fahrenheit. 

T\ = the upper absolute temperature of a pair of temperature-limits. 
T<l = the lower absolute temperature of a pair of temperature-limits. 
t = temperature Fahrenheit. 

V '= volume, in cubic feet unless otherwise stated. 
W— weight, in pounds avoirdupois unless otherwise stated. 
X = the external work involved in the vaporization of dry saturated steam. 
x = the unknown or indefinite exponent of the volume-ratio (in the expan- 
sion of a gas) in the equation connecting it with the pressure-ratio. 
y = the unknown fractional proportion of steam to total mass in a steam- 
and-water mixture. 



PART I 
THEORY 



THE THERMODYNAMICS OF 
HEAT-ENGINES 

CHAPTER I 
THE GENERAL PRINCIPLES OF ENERGETICS 

The term thermodynamics is derived from two Greek 
words : 6ep/JLo<i, meaning " heat," and SiW/u?, meaning 
" force." It refers to the study of the relations between 
heat and mechanical energy. 

There is scarcely a science so widespread in its appli- 
cations as thermodynamics. To the student of modern 
science the source of all things is heat. Wind and weather, 
all vegetation, all life, take their origin and find their 
strength in heat. Not only man and all of the higher 
forms of animal life, but the lowest and simplest of vege- 
table types as well, depend upon heat for their existence 
and absorb it as constantly as their food. This heat- 
energy it is the continuous duty of their lives to convert 
into mechanical motion, or work. In inanimate nature the 
same is true. The vigor of the storm, the irresistible mo- 
tion of the glacier, the erosion of mountains and the filling 
^of valley or lake ; the frosted window-pane, the sculptured 
iceberg, the mirage of the desert, the singing bullet, and 
the quiet beneficent storage of water in pond and reservoir ; 



2 THE THERMODYNAMICS OF HEAT-ENGINES 

the gun, the rocket, the steam-engine and the gas-engine, 
— all are manifestations of heat and its transformations 
into motion, all are more or less thermodynamic in char- 
acter. For the study of each a comprehension of the 
principles of thermodynamics is necessary. Each is an 
illustration of the working of those principles and of the 
laws which limit them. 

The two quantities which form the stock-in-trade of all 
heat-engines — the raw material and the manufactured 
product, so to speak — are heat and mechanical energy. 
Both of these quantities belong under a common classifica- 
tion title as being both of them forms of energy. 

According to the modern scientific analysis of natural 
phenomena, the entire universe is made up of two broad 
subdivisions of existence — matter and energy. The con- 
ception of matter is familiar to all. It apparently appeals 
directly to our senses at every turn. It apparently, too, 
appears in a myriad of different forms. The study of 
chemistry, however, teaches that in all of these infinitely 
varying forms there are really to be found only some 
eighty or ninety distinct kinds, or elements, of matter, such 
as carbon, hydrogen, calcium, etc. Physics, again, tells us 
that each of these elements, and each of their innumerable 
chemical combinations, exists in four distinct conditions : 
solid, liquid, gaseous, and radiant forms of matter. 

The fundamental law which underlies both chemistry 
and physics is that of the Conservation of Matter. This 
law is the statement of the fact that, no matter what physi- 
cal alterations matter may undergo, whether from solid to 
liquid or from liquid to gaseous, or the reverse, and no 
matter what chemical transformation may occur, whether 
the substance be burnt as fuel in a furnace, decomposed as 
lime in a kiln, or corroded with acids, throughout every 



THE GENERAL PRINCIPLES OF ENERGETICS 3 

conceivable change in its form and characteristics the 
quantity of matter in the universe remains always the same. 
Man may change its locality at will. Working in harmony 
with the laws which limit all natural action, he can trans- 
mute its outward appearance in a myriad of different ways, 
to suit almost every imaginable whim or purpose. But to 
alter in the slightest degree the existing quantity of matter, 
to create one particle, or to destroy a single atom, he has 
always been and always will be impotent. The universe is 
eternal. In the face of its steadfast continuity man's mo- 
mentary existence and evanescent will are as cloud-wreaths 
against a mountain-side. 

The other of the two broad subdivisions of all things 
natural includes the various forms of energy. These forms 
of energy are distinguished by external characteristics upon 
which is based a proper classification. But this classifica- 
tion is not parallel with that of either chemistry or physics. 
Whereas, so far as the science of chemistry has yet pro- 
gressed with certainty, the various chemical elements can 
never be confused with one another, the several forms of 
energy are separated by lines often quite indistinct to our 
limited discernment. Whereas one chemical element can 
never be altered into any other, the various forms of 
energy are each of them capable of being transformed, 
under favorable conditions, into any one of the other forms. 
Nevertheless, under and through all study of any of the 
forms of energy, or of their mutual transformations, runs 
one fundamental and immutable law, that of the Conserva- 
tion of Energy, quite parallel to the law of the conservation 
of matter. It states that, no matter what transformations 
may occur to any form of energy, the quantity of energy 
in the universe after the change is quite the same as it 
was before. These transformations may greatly alter its 
appearance to our senses, or its effect upon other things ; 



4 THE THERMODYNAMICS OF HEAT-ENGINES 

yet the rule remains unbroken. No energy has been de- 
stroyed and none created. Energy, the consort of matter 
in that union of all things which we call Nature, is eternal, 
is constant, as is matter. 

These two laws, of the conservation of matter and of 
energy, together constitute the foundation of modern engi- 
neering science. It was ignorance of the law of the con- 
servation of matter which led the alchemists of old on their 
long, costly, hopeless search for the "philosopher's stone," 
by mere touch of which all matter might be transmuted 
into gold. It is ignorance of the law of the conserva- 
tion of energy which has more recently led to the strenu- 
ous, fascinating, but hopeless endeavor for the production 
of " perpetual motion," or mechanical energy produced 
without the aid of any anterior supply of energy. Volumes 
of history could be crowded with. the list of the lives ruined 
or lost in the following of these two dreams, and no more 
forcible illustration could be adduced to prove the indis- 
pensable need of every engineer for a complete and 
accurate grasp of these two laws and of their absolute 
rigidity. 

What is meant by the words energy and transformation ? 

Energy is defined as "the ability to overcome resistance." 
All natural phenomena consist, in one way or another, of 
an overcoming of resistance ; therefore all phenomena 
known to man are manifestations, in one way or another, 
of energy. It is a quality inherent in all matter, to some 
extent. 

Yet, so far as we now know, any or all of the energy 
inherent in any body is quite distinct and entirely detach- 
able from that body. Although human experience has 
never covered the case of a bit of matter possessing no 
energy, yet we always conceive of such a case as quite 



THE GENERAL PRINCIPLES OF ENERGETICS 5 

possible, because some portion of the energy of a mass of 
matter is always removable from it ; nor is there any 
definite limit to the amount removable. Thus, a weight 
lifted above the earth possesses energy, for it is able to 
overcome resistance in falling again toward the earth. 
Yet the energy resident in the body of the weight is quite 
distinct from the matter composing the body ; for if the 
weight be removed to a locality where it no longer possesses 
weight, say to the centre of the earth, the mass of the body 
will remain unchanged, while the energy has entirely dis- 
appeared. Similarly, a moving projectile possesses the 
ability to overcome resistance, and therefore contains 
energy ; but when it has been brought to rest, if not too 
suddenly, the matter will remain unchanged, while the 
energy is gone. 

The overcoming of resistance involves two distinct 
factors: (1) the resistance, and (2) the extent to which it 
is overcome. The first is most simply denned in terms of 
force, a thing so familiar to us all that no further analysis 
or definition of it need here be attempted. It does need a 
unit of measurement, however, and for this will be ex- 
clusively adopted, in the following pages, the pound 
avoirdupois. Secondly, the extent or distance through 
which the overcoming of resistance is effective is another 
matter of familiar experience and needs no definition. 
The unit for its measurement which will be used hereafter 
is the English foot. 

It is next to be noted that the overcoming of resistance 
involves the expenditure of energy. The distance traversed 
by the acting force in the direction of its exertion is the 
same as that through which the resistance is overcome in 
a direction opposite to its exertion. This is the modern 
equivalent to saying that " action and reaction are equal 
and in the opposite direction." The driver must lose as 



6 THE THERMODYNAMICS OF HEAT-ENGINES 

much energy as the driven gains, and the driven must gain 
as much as the driver loses. The distinction between loss 
and gain is an artificial one and depends solely upon the 
point of view from which they are regarded ; they are, 
therefore, properly distinguished by positive and negative 
signs, it being usual to consider the acting force as positive 
and the absorbing resistance as negative. 

From the foregoing considerations is reached the funda- 
mental idea that the unit of quantity of energy is that 
manifested when a force of one pound overcomes a resist- 
ance of one pound through a distance of one foot, during 
which process the acting force has expended and the 
resistance has absorbed one foot-pound of energy. 

Work. — As this particular form of energy has its indi- 
vidual name, in the classification of different forms of 
energy, and is known as mechanical energy, or work, the 
foot-pound is also the unit for measuring quantities of 
work, the word work being understood in its mechanical 
sense. Further, as mechanical energy has always been 
the most familiar of all of the different forms of energy 
and" the one with which acquaintance is first made, it has 
become the custom to base the units of the other forms of 
energy used by the engineer upon the foot-pound, and to 
define them in terms of it as the fundamental unit. 

Power. — Closely associated with the idea of mechanical 
energy is that of power. Power is defined as the rate at 
which work is done. It therefore involves, besides the 
factors of force and distance which are involved in the 
idea of work, the additional factor of time. The unit of 
power is the horse-power, which is defined as that rate 
of doing work which will accomplish 33,000 foot-pounds 
of work if steadily continued for the space of one minute. 
This same rate would accomplish 550 foot-pounds of 
work in one second ; so the latter is often given as a 



THE GENERAL PRINCIPLES OF ENERGETICS 7 

definition of the horse-power. In metrical units the horse- 
power is defined as the rate which would accomplish 
seventy-five kilogram-meters of work if continued for one 
second. 

It is obvious that power, or the rate at which work is 
done, has nothing to do with the quantity of work accom- 
plished. A very slow rate may accomplish a great deal of 
work, if it only keeps at it long enough, while a very high 
rate may be represented by a very small quantity of work, 
provided it be expended in a sufficiently short space of 
time. Thus, a pump driven by a motor having a power of 
much less than one horse-power, by working steadily, day 
and night, for weeks and months, may lift millions of tons 
of water and thereby perform tens of millions of foot- 
pounds of work; yet its power is none the less a fraction 
of a horse-power. Or, on the other hand, a rifle, accom- 
plishing, in imparting motion to the bullet, no more work 
than does a boy in throwing a heavy stone, by performing 
the task in an exceedingly short period of time actually 
does exert a power, or a rate of performance, of something 
like 5000 horse-power. 

The Forms of Energy. — Energy, the eternal, manifests 
itself to the human senses in several distinct aspects, or 
forms. Familiar instances are those of mechanical motion, 
light, sound, electricity, heat, etc. In fact, there is no 
phenomenon under the canopy of heaven which may be 
defined as constituting a process, as being aught else than 
a dead static fact, or condition, which is not a manifesta- 
tion of energy, in some form or another. The motion of 
the sun and its planets, the rise of the tide, the shower of 
rain, the flash of lightning and the peal of thunder, the 
growth of vegetation and the most intricate processes of 
animal life, are all to be classed, in their broadest sense, 



8 THE THERMODYNAMICS OF HEAT-ENGINES 

as manifestations of energy; and all of them must live 
within its laws. 

This myriad of different forms of energy may be sub- 
divided into groups or classes in a variety of ways, each 
based upon some line of distinguishing characteristics. 
Of these the broadest classification is that into three 
main groups, viz. : Kinetic, potential, and vibratory forms of 
energy. 

The term kinetic, from the Greek kivtjtos, " motion," 
always implies the presence of motion. Hence, kinetic 
energy may be defined as that involved in the motion of a 
mass. It is one of the primary conceptions of mechanics 
that energy is always required in order to impart motion or 
change of motion to mass ; and, obversely, that mass can 
never part with its motion except in the course of over- 
coming resistance. As motion without or separate from 
mass is a purely geometric conception, finding no counter- 
part in human experience, it will be sufficient, hereafter, 
to refer to kinetic energy simply as consisting of motion. 
The most familiar instances of kinetic energy are those of 
a moving projectile, of a hammer, or of a fly-wheel. 

The term potential, from the Latin potior, "to be able," 
always implies latent ability. It is, however, the dis- 
tinguishing characteristic of potential energy that it has 
always the appearance of having been acccumulated by 
previous motion, or kinetic energy. It will therefore be 
most clearly defined to the mind as stored motion. This 
fact may be easily fixed in the mind by association of the 
' words potential and position ; for potential energy is always 
involved in mere position, attained as the result of motion, 
as contrasted with kinetic energy or the motion which gave 
it birth. 

To illustrate, the most familiar instance of potential 
energy is that of a weight raised from the earth. By rea- 



THE GENERAL PRINCIPLES OF ENERGETICS 9 

son of its separation from the earth it is able to overcome 
resistance in again falling into contact with the earth. 
The ability to do work is there, latent, unasserted; yet it 
is none the less powerful. Thus, imagine a six-inch rifle 
aimed vertically into the air and fired. The projectile, as 
it leaves the muzzle, possesses an enormous supply of 
energy which it has received from the explosion of the 
powder. As it arises, this kinetic energy is gradually and 
constantly being expended in overcoming the force, or 
resistance, of gravitation. The projectile must finally reach 
an elevation at which, its supply of kinetic energy having 
been all expended, it will have come to rest. 

Let us suppose that a scaffolding had been built up 
from the earth to this point and that a man standing upon 
it should catch the projectile, at the instant of its stop, and 
hold it there. The projectile must contain at this instant, 
if we neglect the friction of the atmosphere, the entire 
amount of energy with which it left the gun ; for if it be 
released, it will exert, in its fall, exactly that amount and 
will reach the earth with as high and destructive a velocity 
as it had at the start. But at the start this energy would 
have blown the man to atoms had he been so unfortunate 
as to come into contact with the projectile. At the top he 
may handle the deadly thing with entire impunity. 

What is the difference ? It is not easily to be explained, 
but a broad fact it certainly is, one which may not yet be 
stated as an absolute law but which is certainly more than 
a widespread coincidence, that the human organism is 
extremely sensitive to kinetic forms of energy but takes 
almost no cognizance, except through the medium of de- 
ductive reasoning, of potential forms. The distinction is 
important, even in the elementary study of engineering 
problems, because there is no one source of engineering 
failure, or of commercial, economic, and political failure, 



10 THE THERMODYNAMICS OF HEAT-ENGINES 

for that matter, so common as a lack of proper estimate 
of the possibilities concealed within the latent, invisible, 
potential forms of energy. 

Of the third group of energy-forms, the vibratory ener- 
gies, little may be said here, in spite of the fact that they 
are the most widespread and familiar of all the forms. 
Vibratory energy consists of periodic alternations between 
kinetic and potential forms of the same kind. The simplest 
instance is that of the pendulum. Here, supposing the 
pendulum to be already in swing, is visible at the centre 
of its arc the kinetic energy of the mass of the pendulum- 
bob. This carries it up the one side of its arc, the kinetic 
energy being gradually transformed into the potential form 
of separation from the earth, until the extreme height of 
swing is reached. Here the transformation is complete 
and the energy all potential in form. But it does not re- 
main so a single instant. The bob has no sooner come to 
rest than it starts upon its return-swing, reconverting its 
potential energy into motion until the lowest point of the 
arc be reached, when the process is repeated up the other 
side! 

Here the alternations between the two forms is slow 
enough, usually, so that the eye, or the ear, as in the case of 
a clock, might easily distinguish the separate beats and count 
them. But if we substitute for the force of gravity the 
elasticity of a spring, and make the pendulum quite short, 
the rapidity of the beats may easily elude the eye. The 
apparatus becomes blurred in appearance, and we know 
the rapidity of vibration only as a degree of smoothness 
of blur. If the vibrations be well above sixteen to the 
second, the ear can no longer count the separate sounds 
emitted by the pendulum in passing the centre ; they be- 
come blurred into a continuous sound, or note, and the 
rapidity of vibration is known only as pitch. Yet each 



THE GENERAL PRINCIPLES OF ENERGETICS 1 1 

vibration is none the less a complete cycle of alternate 
transformations of kinetic into potential energy, of poten- 
tial back into kinetic. 

If we substitute for the gravity-pendulum an absolutely 
fluid mass, to which water is a close simile, we shall find 
that each particle of water near the surface is easily set 
moving in a vertical circular orbit which will carry it alter- 
nately above and below the mean level of static equilibrium. 
Its extremes above and below, the crests and troughs re- 
spectively of the water-wave, correspond to the extremes of 
the pendulum's swing, its energy there being all potential 
in form, either positive or negative. Its alternations be- 
tween the two constitute a new form of vibratory energy. 
Here, however, we find a strange and unwavering tendency 
to transmit the energy from one particle to another, and 
that almost without loss, so that the form of the vibration, 
or wave, as it is called, progresses continuously from one 
locality of the surface of the fluid mass to another. This 
is a faculty possessed not at all, or very slightly, by the 
vibratory motions of single solid bodies, such as the pendu- 
lum. It is a property of all fluids, however, and of all 
elastic, continuous solids, so far as transmission within 
their own confines is concerned. In air, which is elastic 
as well as fluid, these (or very similar) waves are called 
sound. 

If we substitute for these well-known substances the less 
familiar ones of the luminiferous ether and the electric 
atom, the vibrations will take the form of light or electricity. 
If it be light, the number of vibrations, always numbering 
in the hundreds of millions per second, is known to human 
sense only as color. But there are many series of vibra- 
tions in this medium which the eye cannot count, just as 
there, are sounds too high in pitch for the ear to hear. 
Those too slow are known as radiant heat ; those too high 



12 THE THERMODYNAMICS OF HEAT-ENGINES 

are known as actinic rays, and can be perceived and counted 
only by the photographic plate. 

There is another broad method of classification than this 
distinction between motion and stored motion. It is based 
upon the medium in which the energy finds expression, 
upon the particular kind or size of mass which happens 
to be in motion. Thus, we make a very easy and useful 
distinction between light and sound, though both are vibra- 
tory ; or between the latent energy of gunpowder and that 
of a lifted tilt-hammer, though both be potential in form. 
Of the many different kinds of energy-forms thus arising 
the following discussion will make use of only the four 
(together with their distinctions between kinetic and poten- 
tial) which are most commonly met with in the engineering 
of heat-engines, viz. : — ■ 

Mechanical : Kinetic and Potential ; 
Electrical : Kinetic and Potential ; 
Chemical : Potential ; 

Thermal : Kinetic and Potential. 

Mechanical Energy is defined as that occurring in the 
motion or the stored motion, relatively to other similar 
bodies, of bodies of some appreciable size and mass. 
Such is the energy of weights, springs, bodies of water, etc. 

Electrical Energy may be defined as that embodied in 
the relative motion or position of separation of the par- 
ticles of some atomic medium, as yet not well defined 
or Understood, but whose characteristics are so markedly 
distinctive from those of the other forms of energy that no 
more scientific definition is necessary here. The atoms 
embodying this sort of energy are known as electrons. 

Chemical Energy is defined as that inherent in the rela- 
tive stored motion or position of molecules of two or more 



THE GENERAL PRINCIPLES OF ENERGETICS 13 

different chemical natures ; that is, such as that between 
a molecule of carbon and one of oxygen, of hydrochloric 
acid and iron, etc. 

Thermal Energy is defined as that inherent in the mole- 
cules of a body by virtue of their motion and position 
relatively to other molecules of the same chemical nature 
within the same body. 

Concrete illustrations of these several forms may be 
drawn from daily experience, viz. : — 

Mechanical, kinetic : A hammer, a fly-wheel, or a pro- 
jectile. 

Mechanical, potential : A lifted weight, a distorted steel 

spring, or the water of a mill- 
pond. 

Chemical, potential : The energy stored in fuel, in gun- 
powder, or in corrosive acids. 

Electrical, kinetic : An electric current. 

Electrical, potential : An electrical charge, in a Ley den 

jar or similar condenser. 

Thermal, kinetic : The sensible heat or temperature- 

heat of a body, as of red-hot 
iron. 

Thermal, potential : The latent heat involved in a 

change of physical state, as of 
the melting of ice into water 
or of the boiling of water into 
steam. 

It is easily apparent that this list of the forms of energy 
would need to be expanded only to include magnetism, 
light, and sound to make it comprehend almost every pos- 
sible phenomenon occurring on the face of the earth or in 
the surrounding universe. 



14 THE THERMODYNAMICS OF HEAT-ENGINES 

One cannot ponder upon this absolute constancy in 
amount of energy, underlying the most complex altera- 
tions in its external appearance, without being finally im- 
pressed with the fact that the alteration is merely external 
and that the real essence of the energy continues to exist, 
unchanged and uninterrupted ; and that what we regard as 
a change does not occur in the thing observed, but merely 
in its locality and in its relation to the observer. For 
all of these forms of energy consist of motion or of 
stored motion ; and both motion and position are purely 
relative. 

Without any need for an attempt at building upon this 
fact a system of philosophy, it is nevertheless necessary 
that the engineer, of all persons, should realize this fact to 
his finger-tips. Otherwise he will frequently be deceived or 
caught napping. No pantomime actress was ever quicker 
or more capricious in her lightning-changes of costume than 
is energy. It is anything and anywhere. At one moment 
heat in the hundred-thousand-mile flames of the sun's sur- 
face, ninety millions of miles away ; the next moment light 
in a man's eye; the next sealed in his brain as an optic im- 
pression and memory which he carries to his grave. Now 
the cold, black inertness of the coal-seam in the bowels of 
the earth ; soon the glare of the boiler-furnace ; then the 
invisible energy of the silent steam ; a moment after, the 
battle of a mighty ship with the fury of the gale, or the peace- 
ful glow of an electric arc miles away. But the actress re- 
mains the same. No matter what the guise, on the stage 
or in the world the impersonator retains the individuality 
of a continuous existence ; and in the latter case the im- 
personator is always motion, — the motion or the stored 
motion of mass. 

In illustration of this truth instances of such a change of 
costume, from every different one to each of the others, can 



THE GENERAL PRINCIPLES OF ENERGETICS 1 5 

easily be found, not only in the wide field of scientific ex- 
periment but in the much narrower one of the useful arts. 
They are all possible; they all occur in the everyday life 
of the universe ; they are nearly all of them familiar to the 
average observer. Most of them are turned to account in 
the production of wealth. Thus : — 

From Mechanical to Thermal : 

The heating of a body subjected to friction or 
impact. 
From Mechanical to Electrical : 

The dynamo or the glass-plate electric machine. 
From Mechanical to Chemical : 

The setting-off of a detonating compound, such 
as fulminate of mercury or nitroglycerine. 1 
From Electrical to Thermal : 

The heating of any non-conductor by the pas- 
sage of an electrical current. 



1 This is one of the most obscure and unfamiliar of energy-transforma- 
tions. The energy revealed in the decomposition of an explosive, which finds 
expression in heat, light, sound, and violent mechanical movement, is the 
equivalent of the chemical energy originally stored in the explosive ; the 
phenomenon should be classed as a transformation of chemical into one or 
more of these other forms of energy. But every explosive, like a row of blocks 
stood upon end, needs some initial impulse or supply of energy from without 
in order to set in motion the process of energy-transformation. With the 
majority of them, as with gunpowder, this preliminary supply of energy must 
take the form of heat, in the process of ignition. But in many others, such as 
those mentioned in the list, heat will not initiate the explosion ; a sudden jar 
will alone suffice. In this sudden shock to the explosive a minute quantity of 
mechanical energy is absorbed in upsetting the equilibrium of structure of a 
few molecules ; the consequent release of energy from the fall and crash of 
these few into their more stable components upsets the rest, and the explosion 
ensues. This preliminary microscopic transformation of mechanical energy into 
chemical energy is the only instance of its class which has yet occurred to the 
writer ; but it is of considerable importance in the arts, for the detonation of 
the ordinary rifle-cartridge and of most heavy explosives depends upon it. 



1 6 THE THERMODYNAMICS OF HEAT-ENGINES 

From Electrical to Chemical : 

The electrolysis of a chemical compound by the 
passage of current, as in electroplating, or in 
the charging of a storage-battery. 
From Electrical to Mechanical: 

The electric motor; the solenoid. 
From Chemical to Mechanical : 

The evolution of a gas from the combination 
of two solids or liquids. 1 The chemical fire- 
engine. Animal activity. 
From Chemical to Electrical : 

The evolution of an electrical current from a 
primary or secondary battery. 
From Chemical to Thermal : 

The combustion of fuel. Animal heat. 
From Thermal to Mechanical : 

The expansion of a heated body ; the steam- 
engine. 
From Thermal to Electrical : The electropile. 2 
From Thermal to Chemical : 

The lime-kiln ; the growth of vegetation. 

Not only the material arts, but the entire daily life-ex- 
perience of man as well, swarms with a myriad of such 

1 This transformation is a very common but, outside of the processes of 
animal life, not a very obvious one. The mechanical work performed by the 
cold evolution of gas from a combination of solids or liquids is usually only 
that of displacing the atmosphere. It is utilized in the arts, however, in the 
operation of chemical fire-engines and extinguishers. 

2 The electropile is a true representative of this class of transformations, but 
one very narrowly limited in its applications. The invention of an apparatus 
which will perform this transformation upon a large scale, without the inter- 
mediate transformation into heat, as through the medium of a furnace, boiler, 
steam-engine, and dynamo, is one of the unsolved problems of the day which 
attracts inventors to its solution by the most unlimited promise of pecuniary 
reward. 



THE GENERAL PRINCIPLES OF ENERGETICS 1 7 

phenomena, all of which consist of and reveal transforma- 
tions of energy from one outward aspect to another. With 
energy not only is frequent alteration of form possible, it is 
apparently inevitable. Not only do all material phenomena, 
but life itself — anything, in fact, which can be defined as 
an event or occurrence — consists of a series of energy- 
transformations following one another in ceaseless conti- 
nuity. There is, so far as the human mind can grasp, no 
possibility of a stop. With many forms of energy the 
transformations follow one another with a rapidity which 
the mind is entirely unable to comprehend ; with others, 
such as those visible in moving machines, they are more 
deliberate and are easily traced ; with others, such as those 
occurring in the heavenly bodies, the changes are so slow 
or infrequent that human experience is too brief to cover 
and perceive them directly, though their occurrence is 
known by scientific deduction. 

Without doubt, light is one of the most evanescent of all 
forms of energy. Its natural speed of 186,000 miles per 
second carries it across all terrestrial distances in the 
twinkling of an eye, and it cannot meet an opaque body 
without being transformed into heat. Yet the light travel- 
ling from the sun to the earth exists unchanged for over 
eight minutes; that coming from Sirius remains unchanged 
for some twenty years ; while astronomers tell us that 
there are fixed stars, or suns, so distant that their light 
which we perceive to-day must have left its source before 
the birth of Christ and therefore must have existed as light 
for some two thousand years without alteration. Sound, 
too, is short-lived. Though much slower than light in 
transmission, the resistance which it meets in its denser 
medium, air, is so great that only a few minutes may elapse 
before it too becomes heat. Electricity is quite as short- 
lived as light. 



1 8 THE THERMODYNAMICS OF HEAT-ENGINES 

The other forms of energy change less frequently. 
Familiar types show the opposite extreme and apparently 
never 'change. The potential energy stored in the "ever- 
lasting hills" appears to be eternal; but the hills are but 
the creatures and toys of a childish ocean not yet full- 
grown, which will yet be the death of them with its cloud- 
bursts and tidal waves. The aged ocean is itself but the 
first-born of Mother Earth ; she herself is but the baby of 
the solar system ; and even the sun, patriarch and ruler of 
all the planets, sure to outlive in his enormous might the 
biggest of them, is having his age estimated by modern 
astronomers as by his teeth and the probable date of his 
death foretold in serious discussion. There is nothing 
known which is stable. There is nothing which will not, 
eventually, change into something else. 

To the untrained observer this endless procession of 
alterations of form is as a kaleidoscope to the childish eye. 
All is bewildering confusion, — wonderfully beautiful, 
but distracting and leading nowhere. But to one who 
knows that the kaleidoscope consists of nothing more than 
a few bits of colored glass and three mirrors, its explana- 
tion is easy and the mystery is gone. The different bits 
can be identified and utilized to produce new effects at will. 
So with this ceaseless chain of energy-transformations 
visible in nature ; we must first learn to identify the com- 
ponent parts, the several forms of energy. We watch a 
bit of sun-heat grow a tree ; we fell the tree for fuel ; fed 
into a boiler furnace it makes steam ; with the help of an 
engine the steam makes motion, which drives a dynamo, 
etc. At each link of the chain the form of energy must 
first be identified. It then becomes clear that only the 
form, and not the quantity, of the energy has changed. ' 
In each case the senses are so deceived by the complete 
contrast between former and latter appearance, and by 



THE GENERAL PRINCIPLES OF ENERGETICS 1 9 

the apparently complete lack of continuity underlying the 
transition from one to the other, that the mind is with 
difficulty persuaded that there has been no change in 
reality and that the transformation has been, to that extent, 
a deception. Nevertheless, the unalterable fact is as has 
just been stated. The existence of the energy has been 
continuous ; its locality and its effect upon us are what 
have changed. Its amount, its power, its possibilities, are 
just what they were before. And in each alteration of 
form the direction of the step shows no evidence of chance, 
of indecision, of mystery. Each is guided by absolute law, 
— by a law which can be identified, defined, utilized. 

In order to trace this law, attention must next be turned 
to the question of the dimensions of energy. 

The Dimensions of Energy. — If each of these diverse 
forms of energy be examined more closely, it will be per- 
ceived that it does not assert itself to our senses as a 
single simple thing, to be comprehended, as to amount 
and characteristics, at a single effort. Indeed, with all 
except one or two it will appear that neither our senses 
nor any instrument which we can construct will be affected 
by and measure its quantity directly. Thus, in its mechani- 
cal form, energy appeals to us, if potential, merely as a 
force. We cannot directly perceive what amount of motion 
has been stored therein or through what distance that force 
will act if released, though we may reason it out. So 
knowledge as to the force alone is no knowledge as to the 
amount of energy present. If the energy be kinetic me- 
chanical in form, we usually can perceive it as a velocity. 
But the mass concerned is not directly visible, and the 
velocity alone will not measure the energy present. It is 
true that the ballistic pendulum will measure kinetic energy 
directly, and so will the electric watt-meter. But in each 
case the instrument takes cognizance of two factors going 



20 



THE THERMODYNAMICS OF HEAT-ENGINES 



to make up the energy and integrates them accordingly. 
In reality it does within its own mechanism just what must 
be done in the measurement of the quantity of any and 
every form of energy: (i) The measurement of two 
distinct factors, and (2) their mathematical combination 
into a result which is a measure of the quantity of energy. 
In the most familiar case, that of mechanical energy, these 
two factors are force and distance, and in every other form 
of energy the two factors are these same familiar ones 
more or less disguised. A list of them is appended. 





Factor of Intensity 


Factor of Extent 


Form of Fnfrcv 










Name 


Unit of Measurement 


Name 


Unit 


Mechanical, pot. 


Distance. 


Feet. 


Force. 


Pounds. 


Mechanical, kin. 


Velocity. 


Feet-per-second. 


Mass. 


Pounds -H G. 


Electrical, pot. 


Potential. 


Volts. 


Charge. 


Coulombs. 


Electrical, kin. 


Potential. 


Volts. 


Current. 


Amperes. 


Chemical, pot. 


Affinity. 


— 


Mass. 


Molecular 
weight. 


Thermal, kin. 


Temperature. 


Degrees (Fahr.) 
absolute. 


Entropy. 


— 


Thermal, pot. 


Temperature. 


Degrees (Fahr.) 
absolute. 


Entropy. 





In this table the factors for the first five forms of 
energy are familiar to every college student. The factors, 
or dimensions, of thermal energy, however, do not seem to 
be a part of one's everyday experience, nor to be readily 
comprehended as factors of energy. It is for this reason 
that they are placed last in the list. 

The fundamental idea of energy is based upon the com- 
bination of force and distance. In mechanical potential 
energy, where the idea originated, the two factors are 



THE GENERAL PRINCIPLES OF ENERGETICS 21 

plainly visible. What, for instance, is there about the fac- 
tors of kinetic mechanical energy which connects them 
with those of potential mechanical energy, into which it 
transforms itself with the greatest readiness whenever 
opportunity offers ? 

The amount of energy resident in a moving mass is 
known to be expressed by the formula ^ Mv 2 , where M 
is the mass and v the velocity.. But a moving mass is 
known to possess energy only because it will always over- 
come resistance in coming to rest. If it be brought to 
rest by means of a constant negative acceleration in the 
unit of time, one second, it will have traversed the distance 
\ v. If its total energy is to be absorbed in this distance, 
it must be done by a resistance having a mean value of 
^ Mv 2 -4- -jjr v — Mv. But Mv is the force which, if applied 
constantly through the unit of distance to the mass M, will 
produce the velocity v, and hence will, if applied nega- 
tively, destroy the velocity v within the distance \ v. Thus 
kinetic energy is also a function of force and distance. 

In the case of chemical energy the factors of force and 
distance, though not directly visible, are easily compre- 
hensible. The molecules of certain chemicals possess a 
mutual attraction for the molecules of other chemicals 
entirely aside from the mutual gravitation of their masses. 
This attraction is known as chemical affinity. Thus, car- 
bon and oxygen exercise a powerful attraction the one 
for the other. Hydrogen and oxygen do likewise ; but in 
the latter case the affinity between the two is much more 
powerful than with carbon and oxygen. When either 
hydrogen or carbon burns, developing heat, the process is 
nothing more mysterious than a falling-toward-one-another 
of each pair of molecules, and the energy developed is 
exactly analogous to that developed when a weight and 
the earth, mutually attracting one another, fall together. 



22 THE THERMODYNAMICS OF HEAT-ENGINES 

Similarly, when steam is dissociated into its components, 
hydrogen and oxygen, or carbonic acid into carbon and 
oxygen, the molecules require to be separated against their 
mutual attraction. The process is exactly analogous to 
the separation of a weight from the earth against the 
mutual attraction, and the energy absorbed in the dissocia- 
tion is entirely similar to that absorbed in raising the 
weight. In the infinite variety of combinations and dis- 
sociations of chemicals there are as many different degrees 
of affinity and of stages of more or less complete combina- 
tion or separation as there are of different weights and 
levels in the field of mechanical potential energy. 

The cases of electrical and thermal energy are not so 
simple in explanation as the above. Each consists of a 
series of vibrations, in its kinetic form, and of permanent 
separation in its potential form. In the case of heat these 
vibrations and separations concern the molecules of the 
body itself, or portions of them, in terms of their mutual 
cohesion ; in the case of electricity they concern, sup- 
posedly, the electric atoms or electrons. Of the molecular 
construction of matter very little is yet known, so that it 
were unsafe to attempt a too close and accurate descrip- 
tion of the ultimate natures of these phenomena. 

But of this much we can be absolutely sure : If all 
matter, including the electrical, does possess component 
molecules, if these molecules be separated from one 
another by a distance considerable in proportion to their 
diameter, and if they exercise a mutual attraction for one 
another, then, in their vibrations within this attraction they 
must involve alternations of kinetic and potential energy, 
and are to this extent a dynamic parallel to a pendulum or 
to waves in motion upon the surface of a sheet of water, 
requiring a supply of energy from without to be set in 
motion, capable of transmitting that energy with great ease 



THE GENERAL PRINCIPLES OF ENERGETICS 23 

and little loss, and necessarily giving that energy out again 
before rest can again be attained. Similarly, in their static 
separation from one another against their mutual attraction 
they must involve a static parallel to a weight and the 
earth, requiring a supply of energy, or motion, from with- 
out for the performance of their separation, giving little evi- 
dence of the possession of the latent energy so long as the 
separation remains unchanged, and re-developing that same 
energy into visibility when again allowed to come together. 

What is more essentially true than that kinetic energy 
is a disguised function of force and distance, is that poten- 
tial mechanical energy is a disguised function of mass and 
velocity. It will be noted that all of the other items in 
the column of Factors of Extent of which anything definite 
can be said are mere measures of mass. Now mass, when 
the source of the idea is once analyzed, appears plainly 
as that portion or quality of the substance of the universe 
which remains constant. Indeed, it alone is that sub- 
stance. All other features, such as weight, etc., are mere 
attributes, due entirely to the position of the mass in ques- 
tion relatively to other portions of mass. 

The first law of physics, that of the conservation of 
matter, states this fact : — 

" The mass of the universe remains unchanged and is 
unchangeable." 

The definition of mass is the mere obverse of this : — 

" Mass is the unchangeable in matter." 

But the first law of energetics, that of the conservation 
of energy, states that the energy of the universe is un- 
changeable. Since all forms of energy are reducible to 
the motion or the stored motion of mass, and since the 
latter factor has just been defined as a constant, it follows 
that the former factor must also be a constant, when the 
total quantity in the universe is viewed as a unit. 



24 THE THERMODYNAMICS OF HEAT-ENGINES 

The truth of this statement will appear more clearly 
from further consideration of the question. For the pres- 
ent it is sufficient to note that, of the two factors going to 
make up each form of energy, one, of the list which the 
writer prefers to call the " factors of extent," is always a 
function of mass. It, in fact, measures the amount of the 
universe which is involved in the energy in question. The 
other dimension, of the list which he calls the " factors of 
intensity," is always a function of velocity of motion or of 
remoteness of separation of the mass involved. It is a 
function of the mutual relations existing between any two 
or more portions of the mass involved in the first factor. 

It is quite consistent with a conception of the universe 
as a system of portions of mass involving various degrees 
of relative motion and position, that the several masses 
should be regarded as always remaining constant, from 
instant to instant, and that their integrated sum be also 
constant. It is also consistent with such a conception that 
the instantaneous relative motion or position of the indi- 
vidual component masses be ever varying ; indeed, the 
very conception of motion implies it. And yet, if the 
centre of mass of the universe be imagined as fixed, or, 
which is the same thing, if these various motions and posi- 
tions be regarded as purely relative, then the integration 
of their relative velocities and of their degrees of separa- 
tion must always together amount to a constant. 

If, then, to return to the list of energy-factors, the fac- 
tors of the first column, those of intensity, be remembered 
as functions of relative velocity or of relative separation, 
it can be stated that in individual cases they will be ever 
found varying and variable, but in their summation or 
average they will always be found constant in quantity. 

The three basic laws of energetics, approached in this 
way, may be stated as follows : — 



THE GENERAL PRINCIPLES OF ENERGETICS 25 

I. The Conservation of Matter : The mass of the uni- 
verse ever remains constant in quantity. 
II. The Conservation of Motion : The total intensity or 
potential of the universe ever remains constant in 
quantity. 
III. The Conservation of Energy : The total energy of the 
universe, the product of the two foregoing factors, 
ever remains constant in quantity. 

It has already been noted that there exists throughout 
the universe a tendency for every form of energy to trans- 
form itself into some other, and that although this tendency 
is in some cases resisted for periods of time seemingly 
very long, when compared to human standards, yet it al- 
ways finally occurs. It is natural to expect that these 
relative motions and positions of the myriad of portions 
of mass which constitutes the known universe should be 
continually altering themselves into other motions and 
positions. It would seem that there must be some laws 
controlling these transformations, now permitting some 
transformations to take place with instantaneous rapidity 
and now delaying others for ages. 

In the tabular list of page 20 the classification into 
factors of intensity and factors of extent was not made 
primarily upon the basis of the broad theoretic postulates 
enunciated above. It had been observed that the factors 
of energy possess two distinct sets of characteristics. 
Upon these characteristics the classification was based. 
The names intensity and extent were chosen as best rep- 
resenting those characteristics. 1 

1 Mr. George Richmond, in treating of thermal phenomena only, makes 
use of the terms height and breadth instead of intensity and extent respec- 
tively, and there are many reasons why, especially in connection with graphic 
representation of energies by coordinate diagrams, his choice is to be pre- 



26 THE THERMODYNAMICS OF HEAT-ENGINES 

In all forms of energy it is the prime characteristic of 
the factor of intensity that it ever and always tends to a 
decrease and disappearance, by the fall of the energy of 
which it is a part to and through lower and lower degrees 
of intensity, until ultimately the zero of intensity might be 
reached. Thus, taking the most familiar form of energy- 
fall first, mechanical potential energy always tends to 
decrease its factor of intensity, distance of separation, to 
lower and lower degrees of separation until there be no 
separation left. Water, attracted by the earth's gravitation 
and separated from it by evaporation into a rain-cloud, 
tends to fall nearer and nearer to the most intimate pos- 
sible association with the earth, from rain-cloud to hilltop, 
from hilltop into mill-pond, thence to the tail-race, and 
from the tail-race to the sea; and if there were a passage 
from the sea to the centre of the earth, it would flow 
thither. Similarly, a steel spring, stretched by previous 
distortion to a distance of separation from its condition 
of equilibrium, tends always to return, through various 
intermediate degrees of separation, to the position of the 
closest possible approach to equilibrium. 

In a like manner the mechanical kinetic energy of a 
moving body tends always toward a reduction in velocity, 
whether through the medium of friction, of impact, or of 
the imparting of a portion of its energy to other bodies in 
the form of motion. 

Electrical energy, whether kinetic or potential, always 
tends to flow from the point of higher voltage to the point 
of lower, reducing its own potential as it goes. 

In chemical energy the tendency exhibited in reactions 
and combinations is always away from the compounds of 
maximum chemical affinity and minimum chemical stability 

ferred. The writer's reasons for preferring the terms he has adopted are for 
the sake of breadth of definition. 



THE GENERAL PRINCIPLES OF ENERGETICS 27 

to those of less and less intense affinity for one another 
until a level is reached where no further combination is 
possible. 

In thermal energy is exhibited one of the most familiar 
illustrations of all, the constant tendency of heat to flow 
from the body or locality of higher temperature to that of 
a lower temperature. But this illustration does not fully 
cover the law. Heat tends strongly to alter itself to a 
lower temperature even when there is no opportunity for 
flow out of the body in which it lies into another colder one. 
Even if the body be completely isolated from the rest of 
the universe, by an insulation so perfect that both radia- 
tion and conduction are absolutely impossible, its heat will 
alter itself to a lower temperature by other processes when 
opportunity occurs. Discussion of the nature of this oppor- 
tunity must be deferred until later in the argument. 

The converse of this law is also true, viz. : That energy 
never will alter itself spontaneously and without trans- 
formation from a lower to a higher degree of intensity. 
Water will not flow uphill, nor any weight rise from the 
earth, nor a spring distort itself. A moving mass will not 
spontaneously increase its velocity, nor impart its energy 
to a mass moving at a higher velocity. Electricity will 
not flow against a superior potential, nor will a charge 
spark from one jar to another more highly charged. 
Chemical compounds will not combine, unaided, into com- 
pounds having a higher affinity than the originals. Heat 
will never flow spontaneously from a cold body to a hot 
one. 

This being true, how is it that there ever exist any 
higher degrees of intensity, to maintain by their fall the 
current phenomena of the universe from which this law 
has been observed ? It is unimaginable that they should 
all have come from an original store and that the universe 



28 THE THERMODYNAMICS OF HEAT-ENGINES 

is engaged in slowly "running down." It is obvious that 
all of these at first sight unattainable degrees of intensity 
in energy must have been somehow attained, and that 
by means of processes which must be as current and as 
natural now as at any previous time. 

There are two methods by which rise of intensity is ac- 
complished, and only two, and of these the first-mentioned 
is only local and temporary in its effect : — 

(i) There comes from some outside source an additional 
supply of energy to force the apparently unnatural pro- 
cess ; or 

(2) There occurs an intermediate energy-transformation 
into some other form and back again. 

Indeed, it is only so long as the energy remains in its 
original form that the spontaneous tendency to drop in 
intensity is manifested. Thus, water will not flow uphill; 
but it can be pumped up with the aid of some external 
supply of power, or, its potential energy transformed 
into kinetic and back again through the medium of the 
hydraulic ram, it will pump a portion of itself to a 
higher level than the supply. The kinetic energy of 
a moving mass cannot of itself take on a higher velocity ; 
but enforced by an additional supply of energy from 
without it will take on a higher velocity, it will become 
a portion of a larger total quantity of kinetic energy 
having a higher degree of intensity; or, if inherent in an 
elastic body and meeting a body having velocity in the 
opposite direction, it will, by conversion into potential 
energy of distortion and back again, convert a portion of 
itself into a higher velocity. Electricity will not spontane- 
ously flow to a higher voltage ; but its voltage may be in- 
creased by being electrically reenforced ; or it will raise its 
own voltage by means of a double transformation through 
a converter. Similarly, heat will not voluntarily alter itself 



THE GENERAL PRINCIPLES OF ENERGETICS 29 

from a colder to a hotter temperature ; but it may be 
forced to do so by an external source of power acting 
through a refrigerating-machine ; or it may be transformed 
into heat of any degree of temperature by means of a 
double transformation into some other form of energy, as 
into electricity, and then back again by way of the electric 
arc. 

It is especially to be noted, in considering this question 
as to how the at first sight apparently unattainable degrees 
of intensity originated, that in the preceding illustrations 
reliance was placed upon the two methods of increase of 
intensity, viz. : — 

(1) The simple addition of energy from some other 

portion of the universe ; 

(2) The transformation, into the form under discussion, 

of some other form of energy already existent. 

It next appears that whenever the latter method prevails, 
whenever such an energy-transformation occurs, there ap- 
pears to be no broad natural limit to the degree of intensity 
of the secondary form of energy. Thus, there is no broad 
law connecting the muzzle-velocity of a projectile with the 
chemical affinity of the gunpowder. There is a relation 
between the two, but it depends upon the relative mass of 
powder and ball; the relative diameter and length of gun- 
barrel, etc., etc. These are all accidental circumstances, 
not broad principles. The attainment of the highest 
muzzle-velocities has not resulted from the discovery of 
explosives of extreme chemical affinity. Similarly, in the 
electric arc, the temperature attained is not a basic function 
of the voltage producing the arc. Very high voltages 
apparently produce no higher temperatures than quite low 
ones may do, under favorable conditions. In any such- 
case the intensity of the secondary form of energy appears 



30 THE THERMODYNAMICS OF HEAT-ENGINES 

to depend upon circumstances which may be called acci- 
dental in their character, and not at all upon the intensity 
of the primary form. It is plain that the phenomenon is 
such as to permit the attainment, in the secondary form of 
energy, of the highest degrees of intensity known to man, 
and that from comparatively low degrees of intensity in 
the primary form. 

Upon further inspection it appears that these two pro- 
cesses by means of which higher degrees of intensity are 
attained are in every way the obverse of those by which 
intensity lowers itself. For when the energy of a body 
obeys the familiar tendency to fall in intensity, its quantity 
must always decrease. But, since annihilation of energy 
is impossible, decrease is possible only in one of two 
ways : — 

(i) Dissipation, or the distribution of energy to other 
masses of similar character ; 

(2) Conversion into other forms of energy, or the dis- 
tribution of energy to other masses of dissimilar 
character ; 

and these two methods are plainly merely the obverse of 
the two which are listed on the preceding page. Thus, 
the energy of a moving mass of appreciable size, such as 
that of a projectile, always exhibits a tendency to decrease 
its intensity, or in other words its velocity, of motion. It 
will always do this in one of two ways : — 

(a) By meeting other bodies which are perfectly elastic 
and which will partake of the motion of the original mass, 
the result being a greater total mass in motion of a lower 
average intensity ; or (&) by meeting other bodies which 
are more or less inelastic, in which case friction and impact 
occur ; and friction and impact are merely names for the 
conversion of mechanical energy into heat. 



THE GENERAL PRINCIPLES OF ENERGETICS 3 1 

Similarly, chemical energy will always, when permitted 
to free itself, either dissociate a larger mass than its own 
to a lower degree of chemical affinity, or it will convert 
itself into heat, light, electricity, sound, etc., which are 
merely the motions or positions of masses dissimilar to 
chemical mass. 

Electrical energy will always fall in potential when per- 
mitted, and in so doing will convert itself into either heat, 
light, sound, chemical energy, or mechanical motion ; or, if 
in static form and free to conduct itself to a larger mass 
of surface, it will do so, but with a corresponding reduction 
in potential. 

Heat, in the same way, will always tend to alter itself to 
a lower intensity, and in doing so either other masses of 
matter will be heated to temperatures lower than the origi- 
nal or new forms of energy, such as chemical dissociation 
or mechanical motion, will be produced. 

If attention then be turned to the secondary bodies 
which are affected by these processes, it will be plain that 
they are undergoing, in each case, the obverse of the pro- 
cesses under discussion. That is, the intensity of their 
energy is being raised by one of two processes : — 

(a) By the simple addition of energy from the primary 

mass ; 
{b) By the conversion of the primary form of energy 

into their own. 

These three statements of the two processes, on pages 
29, 30, and 31, show them to be identical; on each page 
they were approached from a different point of view. 

Finally, if it be remembered that all forms of energy 
consist of the motion or the stored motion of masses of 
one sort or size or another, it becomes plain that even the 
conversion of energy into other forms is merely a transfer 



32 THE THERMODYNAMICS OF HEAT-ENGINES 

of motion from one mass to another, but in this case to a 
mass of a different kind. Thus, when the energy of a 
moving mass of appreciable size is converted into heat by 
friction, the parallel simultaneous motion of the molecules 
forming the body is altered either into other sorts of motion 
of those same molecules, in so far as the moving body 
itself is heated, or into varied motions of the molecules 
of the bodies with which it comes in contact. Similarly, 
the production of electricity or chemical dissociation from 
mechanical energy is merely the transfer of the parallel 
motions of the particles of the original body into various 
peculiar motions or separations of the numerous masses of 
electrical or chemical atoms of the bodies affected. These 
bodies may remain stationary in relation to the surround- 
ing universe, but they are in possession of active internal 
motion ; and the original source of energy has been brought 
to visible rest. 

Summary 

The universe, viewed in this light, reduces to the follow- 
ing simple statement, which includes all known phe- 
nomena, whether simple or complex, whether inanimate or 
animate, from the motions of the heavenly bodies to the 
action of dust floating in the sunlight, from the quiescence 
of the clod to the most intricate and delicate processes of 
human life and thought, viz. : — 

An infinite series of portions of mass, of infinite variety 
of size and form, linked together by mutual gravitational 
attraction, each associated with the others in a certain 
degree of separation and possessing a certain amount of 
motion. Both separation and motion are purely relative. 
Motion cannot exist without accomplishing separation, 
positive or negative. Separation cannot exist, appear, or 
disappear, without accomplishing motion, positive or nega- 



THE GENERAL PRINCIPLES OF ENERGETICS 33 

tive. Motion, therefore, cannot exist without causing 
change in both separation and motion. The same is true 
of separation. 

Newton's " First Law of Motion " is in no conflict with 
the above statement, because the former, in its application 
to a given mass, assumes the non-existence of all other 
mass. The law stated in the preceding paragraph assumes 
in its premises the existence of all other mass, — not 
merely a few portions of it, but an infinity of portions, 
extending to unthinkable limits. 

Motion and separation are purely relative and are inter- 
changeable. They together constitute the intensity of the 
energy of the universe. The mass embodying the motion 
or separation constitutes the extent of the energy of the 
universe. Both intensity and extent ever remain constant 
in quantity. 

The intensity of the energy of a given mass always 
tends to decrease. When that energy consists of separa- 
tion, the separation tends to disappear, producing motion; 
when it consists of motion, the motion tends to disappear, 
producing separation. 

When the motion is transferred from one portion of 
mass to another (which constitutes what is known as a 
"transformation " of energy), the proportion of intensity of 
motion or separation to extent of mass involved in the new 
form is independent of the degree of intensity in the old ; 
the degree of intensity in the new is always a maximum 
controlled by environment. 

The laws which govern the when and how of such con- 
versions, and which dictate the limit which has been 
cloaked under the expression " the maximum permitted by 
environment," are of the most intricate. Their discovery 
and statement are the work of all the existing sciences and 
of all those yet to be constructed, in combination. 



34 THE THERMODYNAMICS OF HEAT-ENGINES 

These principles may be stated in more concrete and 
popular language in the following form. 

The primary law of energetics is that of the 

Conservation of Energy 

" Energy is eternal. It can never be either destroyed or 
created ; whenever one form of energy disappears it must 
reappear in another form in equal quantity." 

This law comprises two minor principles : — 

I. The Conservation of Mass 
Matter is eternal. It can never be destroyed or created; 
only the mutual relation of its parts can be altered. 

II. The Conservation of Motion 
Motion is eternal. The total amount and the average 
intensity of the motion or the separation (the " stored 
motion ") of the universe remains ever constant and can 
never be destroyed or created ; it can be altered only in 
the mutual relations of its parts. 

Mechanical energy and thermal energy, by virtue of 
their existence as energy, come within the above law as 
special illustrations. Hence the primary law of thermo- 
dynamics, the so-called 

First Law of Thermodynamics 
Heat can be neither destroyed nor created. When con- 
verted into motion or derived from motion, the energy 
disappearing is always the equivalent of that reappearing. 1 
The second great law of energetics is that of the 

1 Of the two minor principles noted above only the first applies in the 
study of thermodynamics, and it is of passive rather than of active interest. 
The second, which covers all the motion of the universe, naturally cannot 
apply to the one peculiar portion of that motion which we call heat. Any 
one portion of the motion of the universe, such as heat-motion, may decrease 
or increase ; it is the sum total only which remains constant. 



THE GENERAL PRINCIPLES OF ENERGETICS 35 

Interchange of Energy 

The form of energy is fickle and evanescent. Neither 
motion nor separation can exist in mass without alteration, 
either into the other one of the pair or into the motion 
or separation of other portions of mass. This alteration 
always takes place from a condition or locality of higher 
to one of lower intensity, and never the reverse. 

Thermal energy, in obedience to this generalization, 
presents the 

Second Law of Thermodynamics 

Heat is fickle and evanescent. It ever tends to alter 
itself from higher to lower temperatures whenever oppor- 
tunity offers, either by dissipation or by conversion into 
other forms of energy. It will never spontaneously alter 
itself from a lower to a higher temperature, but may be 
forced to do so in either of two ways, viz. : — 

(1) By a supply of energy from without; 

(2) By a double intermediate transformation into some 
other form of energy and back again. 

When the tendency of heat to fall in temperature was 
first noted and formulated into a law, it was also noted that 
its dissipation amongst larger and larger masses was the 
commonest method of temperature-fall. No return of the 
heat to higher temperatures seems to have been noticed, 
although it is occurring upon every hand. The situation 
was therefore covered by the so-called " law of the dissipa- 
tion of energy," viz. : — 

The entropy of the world tends to a maximum and the 
temperature to a minimum. 

The modern statement of both sides of the law, in 
similar language, would be : — 

The total entropy and the average temperature of the 
universe remain constant. 



36 THE THERMODYNAMICS OF HEAT-ENGINES 

'■5 
But no such law may probably be true, for entropy and 

temperature are respectively the mass and the intensity of 
only one sort of energy. Since there is as yet no evidence 
accumulated which reveals any fixity of proportion between 
the several sorts of energy in the universe, although in all 
probability that proportion is rigidly controlled by natural 
law, no law similar to the above may be stated which con- 
fines itself to a single form of energy, such as heat. 



CHAPTER II 

THE CYCLE 

The tendency of energy to fall in intensity, whatever be 
the form of energy concerned, almost always results in the 
more or less complete transformation of the energy into 
some other form. But in each of these cases of transfor- 
mation it is to be noted that the continuity of transformation 
is limited. Thus in the case of a weight suspended above 
the earth, it may continue to fall and to convert its energy 
into a kinetic form only until it reach the earth. That once 
reached its potential energy is, in one sense, all gone ; its 
intensity of separation is simultaneously all gone, and the 
progress of conversion must cease. The matter which had 
recently served as a medium for the desired transformation 
of energy is no longer useful for that purpose. 

This fact is a serious obstacle in the practical application 
of the process in the arts. A supply of energy must be 
continuous in order to be useful. Further, it is not ordi- 
narily possible to look to a continuous, inexhaustible supply 
of matter containing the particular form of energy available 
for conversion. We must convert an indefinite amount of 
energy from the available to the desired form by means of 
a finite amount of matter, which must be utilized and relied 
upon as a carrier for the energy. The only known method 
by which such a carrier can perform continuous energy- 
transformation is by endless repetition of the process. 
Thus, water endlessly repeats the process of fall from the 
' rain-clouds to the mill-pond and thence to the sea, and of 
evaporation from the sea to the cloud again. 

37 



38 THE THERMODYNAMICS OF HEAT-ENGINES 

The Working-substance. — The particular form of mat- 
ter used as a carrier in any such repeated conversion is 
called the working-substance.- The repeated process by 
which it effects a continuous conversion of energy is called 
a cycle. The latter may be defined as follows : — 

A cycle is a series of processes, usually four in number, 
by the repetition of which a finite amount of matter may 
effect a continuous or infinite conversion of energy from one 
form to another. 

It is in the application of thermodynamics to the design 
of heat-engines that the study of the form of cycle involved 
becomes of prime importance, much more so than in most 
other classes of energy-conversion. But as the conception 
of heat as a. form of energy, and particularly as its two 
dimensions, temperature and entropy, are more difficult of 
comprehension than is the case with some more familiar 
and concrete forms of energy, the study of the thermo- 
dynamic cycle will be presented in the following pages by 
a running parallel between it and a similar conversion of 
potential mechanical energy, as of stored water-power, into 
some other form, and of heat-energy into some mechanical 
form. 

In every such case the primary form of energy is com- 
posed of two factors, viz. : height and weight in the first 
case, temperature and entropy in the second. If the weight 
of a given mass be assumed to be constant over all ordi- 
nary variations in height, which is very nearly true, it 
would follow that the potential energy involved in the 
weight was equal to the height multiplied by the weight. 
In fact, in all hydraulic engineering we rely upon the for- 
mula that E = H x W, where E is the energy, H the 
height, and J^the weight. But this formula is based upon 
the assumption that all changes of weight take place at a 



THE CYCLE 39 

constant fixed height, and is therefore special in its char- 
acter. To state a general equation based upon the same 
idea, recourse must be had to the calculus. The equation 

then becomes .„ rr;TI7 , N 

dE — HdW. (1) 

This is the fundamental equation of potential mechanical 
energy. 

In the case of heat-energy the situation is not quite so 
simple; at least it is not so apparent. Temperature has 
been defined as some function of the square of the velocity 
of translation of the molecules composing a mass of matter. 
But entropy has not yet been defined to the comprehen- 
sion with even this degree of accuracy of conception. Its 
origin arose from the fact that the early founders of the 

science of thermodynamics found that in any alteration in 

rid 
the quantity of heat in a body the ratio -^-, wherein Q is 

quantity of heat and T the absolute temperature at which 
the alteration takes place, was a very frequently recurring 
and a convenient and useful function for a variety of cal- 
culations. They therefore gave it a name, entropy. The 
fundamental equation of all thermodynamic calculation 
therefore is , n 

dN=*fA, (2) 

where N is entropy, Q the quantity of heat involved in the 
change, and T the absolute temperature at which the change 
occurs. 

This equation gives the mathematical definition of en- 
tropy. Its definition in general terms is : — 

Entropy is that quality of a body which increases when 
heat is added to the body, which decreases only when heat is 
abstracted, and consequently remains constant only when 
heat is neither added nor abstracted. 



40 THE THERMODYNAMICS OF HEAT-ENGINES 

Equation I easily converts itself into dQ = TdN, and in 
this equation we see great similarity to the equation con- 
necting the two dimensions of potential mechanical energy, 
just given with water-power as an illustration. It is the 
integration of this equation, or 






TdN, (3) 



which gave rise to Zeuner's name for entropy, heat-weight ; 
for in the energy developed by the fall of heat from the 
condition T X N X to the condition T 2 N 2 , as given by the 
above equation, the entropy plays a very close parallel to 
the function of weight in the energy developed by the fall 
of a mass from a condition H x W 1 to a condition H 2 W 2 . 

These two definitions, the mathematical and the general, 
must suffice for the identification of entropy with mathe- 
matical accuracy. 

But an approximate conception of its nature which is 
much more useful than the foregoing is to be had from the 
table of dimensions of energy given on page 21. It will 
become very clear, as the study of thermal phenomena 
proceeds, that entropy belongs in the column of factors of 
extent. When a thermic body is left thermally isolated, its 
entropy never alters in quantity, as its temperature may. 
When the body meets another and exchanges energy of 
like sort, there appears no law of connection between the 
direction of the interchange and the degree of entropy 
present. In all such ways entropy shows none of the 
characteristics which appear so plainly as the distinguish- 
ing characteristics of the factors of intensity and which 
form the basis of the Second Law of Energetics. Clearly 
entropy is a dimension of extent. 1 

1 As to the idea frequently promulgated at present in a manner far too 
dogmatic to harmonize with our meagre knowledge of the truth, viz. : that 



THE CYCLE 4 1 

If so, entropy must be either mass itself or else some 
close function of mass. There are obstacles in the way of 
further definition of the first hypothesis ; there is no basis 
which visibly directs one to a determination of the function 
referred to in the second hypothesis. So that further elab- 
oration of the definition seems impossible. Yet the original 
statement remains unshaken. 

It will develop, upon the most superficial inspection of 
thermal phenomena, that entropy is an active variable. If 
entropy be mass, the inevitable deduction must be, since 
the mass of a body remains constant during heating or 
cooling, that the proportion of the mass of the body which 
takes part in thermal vibration must be variable. Since 
thermal energy is normally distributed evenly throughout 
the body containing it, this must mean that there is a vari- 
able portion of the mass of each atom which enters into the 
motion which constitutes thermal energy. This variable 
portion, according to the first hypothesis given above, is 
the entropy of the body. If not, it is at least an active 
factor in that entropy. 1 

entropy is a mere mathematical ratio, heat divided by temperature, it is a 
brief task to point out its untenability. Heat is energy, an eternal, inde- 
structible reality. Temperature is a mere dimension, quite destructible, of 
that reality. Their natures are totally incompatible. There is no more possi- 
bility of a ratio between heat and temperature being nothing more than a 
ratio than there is of a ratio between area and length being nothing else. 

Nor does the hypothesis that temperature is itself energy (based solely 
upon assumptions which apply only to a single form of matter, the gaseous) 
help the situation. Both the isothermal and the constant-heat processes (see 
pp. 47 and 105) deny the tenability of the hypothesis. In the first, energy 
is absorbed with no increase in temperature; in the second, temperature dis- 
appears with no reappearance of temperature. Both processes show an irrecon- 
cilable conflict between the hypothesis and the First Law of Thermodynamics. 

1 This hypothesis is in no way untenable in the face of the present aspect of 
molecular physics. The word atom has completely lost its pristine significance 
of something akin to a microscopic billiard-ball. The atom has never been 
anything more than a concept, based upon certain assumptions in order to 



42 THE THERMODYNAMICS OF HEAT-ENGINES 

Let the two pairs of factors which go to measure energy- 
quantities of the potential-mechanical and the thermal types 
respectively be represented by ordinates and abscissae of 
two respective sets of rectilinear coordinates, the former as 
in Fig. i and the latter as in Fig. 2. Let O W be an axis 
upon which is measured weight, OH an axis on which is 
measured heig]it, OT an axis on which is measured (abso- 
lute) temperature, and ON an axis of entropy. 

Let us suppose that the point A, in Fig. 1, represents 
the condition of a mass having the height of separation 
from the earth of H 1 and the weight W 1 ; then its inherent 
potential-mechanical energy must be E = H x W v Similarly, 
we may suppose that the point A of Fig. 2 represents the 
thermal energy of a mass possessing an absolute tempera- 
ture T x and an amount of entropy N x ; but we cannot simi- 
larly assume that its total thermal energy is Q — T X N V 

clarify certain obscure phenomena. These assumptions have had to grow in 
complexity with the growth of knowledge, until it is now accepted that what 
we attempt to define as an atom is merely a mass of matter very much smaller 
than human dimensions, but which, instead of being an indivisible, elementary 
sort of thing, may be and is probably as complex within itself as are the 
largest known masses of matter. So that it is quite feasible to suppose that 
varying portions of its mass may enter into those motions and positions which 
we lump under the title of thermal energy. 

Moreover, it happens, rather unfortunately, that practically all of the 
brilliant investigations of the mathematical limits of molecular action have 
included in the premises the assumption that the mass involved was constant. 
The sole justification of this is the familiar fact that heating or cooling does 
not alter the total weight of a body. 

For this reason, too, the hypothesis that entropy is merely mass has not 
progressed in limitation. It has neither been proved nor disproved. To the 
student who does not find that it aids his comprehension, complete progress 
through mathematical thermodynamics may be made by means of the two 
exact definitions of entropy already given ; but it will be purely mnemonic in 
its character and correspondingly limited in usefulness. To the student who 
takes the trouble to study the real significance of thermodynamic phenomena 
this hypothesis that entropy is thermal mass, although not demonstrably exact, 
will be found to be a great and proper aid. 



THE CYCLE 43 

For of mechanical energy it is known that it must always 
equal distance times force, but of thermal energy it is only 
true that dQ = TdN. 

In Fig. 1 suppose that the weight at A be released from 
its position. It will immediately fall toward the earth, or 
along the vertical straight line AB. Suppose that at B 
further fall be impossible, by reason of some practical 
limitation. The weight, now being useless for the further 
development of power, must be gotten rid of. If it be 
water, it is easily poured away, as into a tail-race. In fact, 
the processes depicted in Fig. 1 are most clearly imaginable 
as being carried out by a water-wheel made up of an endless 
chain of buckets passing over four pulleys, one at each 
corner of the rectangle. Each increment of departure from 
the axis OH should then be understood as involving a cor- 
responding increase in weight, as of water poured into the 
buckets ; each approach toward OH should be taken to 
indicate a proportionate decrease in weight, as of water 
poured out of the buckets. In this sense the line BC rep- 
resents the emptying of the buckets at the tail-race level. 

It may happen, from practical limitations, that when the 
point C is reached no more weight can be disposed of. 
Either no more water will flow out ; or H 2 C may be taken 
as measuring the weight of the empty buckets themselves. 
If so, this weight must be raised to the Z/j-level before 
more water can be taken in for the performance of more 
work. So the weight H 2 C is lifted along the line CD to D. 

The empty buckets can now be filled, at head-race level, 
along the line DA, after which the process may be repeated 
indefinitely. 

In this cycle the fall of the weight W 1 along the path AB 
develops the work W-^H X — H 2 ). In other words, the 
secondary form of energy is abstracted, after formation at 
the expense of the primary. Along BC no work is done. 



44 



THE THERMODYNAMICS OF HEAT-ENGINES 



Instead, energy of the primary form is abstracted. Along 
CD the negative work W 2 (H 2 — H^) is done upon the work- 
ing-substance by some exterior source of power, or, in other 
words, the secondary form is supplied to the system, being 
at the same time transformed back into the primary form. 
Along DA no work is done, positive or negative. Instead, 
the primary form is supplied. {DA is most naturally taken 




Fig. i 



as the initial process of the four.) Therefore the total net 
work done by the working-substance must be the algebraic 
sum of the two quantities visible along AB and CD, or 



Work=(^- W^(H X -H^ 



(4) 



But in this expression the first parenthesis is the width of 
the rectangle DABC, and the second is the height. There- 
fore, in a rectangular cycle such as this the net quantity of 
energy transformed is measured by the area enclosed. 



THE CYCLE 



45 



It is evident that this same result could be reached by 
subtracting from the primary energy entering the system 
along DA that rejected along BC. The difference, which 
disappears during the cycle, must, according to the First 
Law of Energetics, reappear in the secondary form. 

If attention be turned from this cycle of finite dimen- 
sions to the infinitesimal rectangular cycle having sides of 




Fig. 2 



lengths dW and dH respectively, shown in the centre of 
Fig. i, it is plain that this argument must equally apply; 
or, in other words, the work done in this infinitesimal cycle 
must be measured by its area, given by the expression 



d(dWork) = dWdH. 



(5) 



If this equation be true, the sum of any number of such 
equations must be true ; that is, the equation may be inte- 
grated. Such integration will result graphically, from the 



46 THE THERMODYNAMICS OF HEAT-ENGINES 

second member of the equation, in the development of a 
closed area, its form depending upon the particular func- 
tions existing between W and H. It will result physically, 
from the first member of the equation, in the net total of 
energy transformed from the primary to the secondary form 
by the summation of rectangular infinitesimal cycles ; and 
the physical significance of this summation of cycles, since 
their contiguous sides must always represent two identical 
processes in reverse directions and hence cancelling each 
other, is the same as that of a series of processes repre- 
sented by the free sides. 

It is plain that any closed area may be considered as the 
summation of such infinitesimal rectangular cycles. Hence 
the conclusion is drawn that — 

If in any series of height-weight processes forming a closed 
cycle {that is, one bringing the system back to its original 
position and weight) these processes be represented by lines 
forming a closed diagram, it must be true that the work 
done during the cycle will be given by the integration, to the 
proper limits, of the eqtiation 



Work 



=ffdWdU. (6) 



But this integration will also result in a measure of the 
area enclosed by the diagram, from which may be drawn 
the proof that in any height-weight diagram the work done 
by a closed cycle is measured by the area enclosed by the 
figure depicting that cycle. 

If attention be returned to Fig. 2, it will be obvious that 
the rectangle A BCD represents no such familiar series of 
processes as does that of Fig. I. Nevertheless, it is just 
as true to life and just as frequent in occurrence ; but it 
is much more obscure. If heat of the condition depicted 
at A (Fig. 2) be given the opportunity, it will fall in tem- 



THE CYCLE 47 

perature down the line AB just as promptly and just as 
accurately as will water at A (Fig. i) fall down the plumb- 
line AB ; and in falling it will just as surely develop work. 
At B (Fig. 2) it may have reached the temperature of sur- 
rounding objects. If so, it will be prevented thereby from 
falling farther, at any rate usefully ; just as the water- weight 
at B had reached a point where it could fall no farther 
advantageously. It must therefore be rejected. 

This is supposedly done along the line BC. The line 
BC therefore indicates the abstraction of heat at constant 
temperature, just as in Fig. i it indicated the abstraction 
of weight-energy at constant level. 

At C it may be supposed that no more heat can be dis- 
posed of. If so, the rest must be lifted again, up the line 
CD, to the original temperature-level, by an expenditure of 
energy brought in from without. The heat-carrier, what- 
ever it may be, is then ready to receive another charge of 
heat along the line DA and to repeat the process indefi- 
nitely. This addition of heat necessarily implies increase 
in entropy. The increase is here supposed to take place 
at constant temperature. 

In the case of the water-power of Fig. i, it is compara- 
tively easy to follow with the imagination the several pro- 
cesses of taking in weight, letting it fall, emptying the 
residue, and returning the empty bucket. In the case of 
the heat-engine of Fig. 2, however, the processes are not 
so easy to follow. They are entirely unfamiliar, and it is 
only with the help of the water-wheel simile and the other 
analogies which have been presented that any useful com- 
prehension of them may be had. 

Isothermals. — The processes DA and BC are called 
isothermals, i.e. occurring at constant temperature. They 
necessarily involve, since the entropy increases or de- 
creases, the addition or abstraction of heat. The most 



48 THE THERMODYNAMICS OF HEAT-ENGINES 

familiar instances of isothermal action are found in the 
melting of ice or the vaporization of water under constant 
pressure. In each case large quantities of heat enter or 
leave the body, yet the temperature remains unchanged. 

Adiabatics. 1 — The processes AB and CD are called adia- 
batics. Their location upon the diagram, the entropy 
remaining constant while the temperature alters, expresses 
graphically what the definition of the adiabatic does in 
words : An alteration in temperature without addition or 
abstraction of heat, or, in other words, while completely 
isolated from heat-exchange with the surrounding universe. 

There lies herein a seeming paradox which must be 
cleared from the student's mind before further progress 
may be made, viz. : If heat be neither added to nor 
abstracted from the body, how can the temperature of the 
body, and also (as is clear from the diagram) its quantity 
of heat, be altered ? The answer is : By conversion into 
work. During the adiabatic fall of heat no heat is lost as 
heat to the surrounding universe ; but heat is steadily lost 
by conversion into the form of work. There is no heat 
abstracted ; but there is heat lost, for work has been 
abstracted. 

Similarly, during adiabatic rise of heat in temperature 
no heat is added to the body; but heat is gained, in the 
form of work, from the outside universe. In the adiabatic 
processes there is no exchange of heat between the body 
and its surroundings ; there is an exchange of energy, 
however. 

The process may be perfectly paralleled by a man who 
goes down town with his pockets full of money. He may 
not have been robbed of a cent, yet he may return with 
his pockets empty ; but the supposition is that his arms are 
full of bundles which he has purchased. Not a cent has 

1 The word adiabatic signifies " without exchange." 



THE CYCLE 



49 



been abstracted from him, yet he has ceased to possess his 
money; it has been transformed into another form of 
economic energy. 

The adiabatic process may be said, in one very true 
sense, to be the only natural heat-process, for it is the only 
one in which the body acts freely and spontaneously of 
itself, without thermal influence from without. It is the 
process which every quantity of heat would follow if left 
to itself. Hence it is of prime importance in thermody- 
namic work as being the process from which all heat-pro- 
cesses depart only so far as they are forced to do so by 
purely external and often artificial forces. The adiabatic 
fall of temperature, which every portion of heat spontane- 
ously undertakes when permitted to do so, is as natural a 
specimen of gravitation as that which called Newton's 
attention to the force which holds the universe together. 
The adiabatic is the plumb-line of heat-energy. 

In this cycle of heat-processes the quantity of primary 
energy or heat supplied along the path DA can be found 
by the integration of Equation 3. Since in this case 2" is 
a constant, T v the integration becomes 

Qx=T 1 {N,-N^. (7) 

Similarly, the quantity of heat extracted along BC becomes 

Qi=T^N x -Ni\ (8) 

Subtracting, according to the First Law of Thermody- 
namics, the total quantity of secondary energy, or work, 
appearing as the result of the complete cycle must be 

Q w = 0i - <2 2 = ( T x - T^N, - N 2 ). (9) 

For along AB and CD no primary energy is added to or 
abstracted from the system. 

But in this expression the first parenthesis is the width 
of the rectangle DABC and the second is the height. 



50 



THE THERMODYNAMICS OF HEAT-ENGINES 



Therefore, in a rectangular entropy-temperature cycle 
such as this, the work done is measured by the area 
enclosed. 

From this idea can be developed the proof that — 

In any closed entropy-temperature cycle the quantity of heat 

transformed into work is measured by the area enclosed, in 

a manner similar to that used with the height-weight cycle. 

In this case, however, it is important to establish one of the 




FIG. 3 



intermediate steps as a general principle. This can be 
done by means of Fig. 3. 

In Fig. 3, let AB be any heat-process whatever. Equa- 
tion 2, transformed, gives 



dQ = TdN. 



(10) 



The right-hand member of this equation is represented 
geometrically by the elementary vertical strip having the 



THE CYCLE 5 1 

width dN and the height T. The integration of the above 
equation to any limits, such as T Q and T x or N and N v 
gives, algebraically, Q, the quantity of heat involved in the 
process, and geometrically, the area between the curve AB 
and the zero-axis of temperature and between the limiting 
ordinates, or the area N Q CDN V Since AB is any curve 
and the limits chosen were any limits, the following law 
may be deduced : — 

If any heat-process be depicted by a curve in an entropy- 
temperature field of coordinates, the heat handled during the 
process is measured by the area between the curve, the zero- 
axis of temperature, and the limiting ordinates. 

T 




Fig. 4 



If attention be next turned to any closed diagram on 
an entropy-temperature field of coordinates, such as that 
shown in Fig. 4, it is plain that it may be considered as 
made up of two portions, viz. : — 



52 THE THERMODYNAMICS OF HEAT-ENGINES 

(i) The portion ABC in which the entropy changes 
from a minimum at A to a maximum at C. This portion 
must indicate the addition of heat to the system. This 
quantity of heat supplied will hereafter be referred to by 
the symbol Q v 

(2) The portion CDA during which heat is abstracted 
from the system. The quantity of heat abstracted will be 
represented by the symbol Q 2 . 

By the preceding argument, Q 1 is measured by the area 
aABCc. Similarly, Q 2 is measured by the area cCDAa. 
The heat converted into work must be their difference, or 

Qw = Qi - £2 = the area ABCD, (n) 

Since Fig. 4 represents any closed diagram, the argu- 
ment is a general one. 

Therefore, if any closed cycle of heat-processes be repre- 
sented on the entropy-temperature field of coordinates by a 
closed figure, the heat converted into work at each revolution 
of the cycle is measured by the area enclosed. 

Conditions Essential to the Production of a Cycle. — The 

truth of the following statement of the conditions which 
must exist before a closed cycle such as that just discussed 
may come into existence will be obvious from the preceding 
argument. The original statement, made in reference ^o 
thermodynamic cycles only, was defined so long ago as 
1824, by Sadi Carnot, then a lieutenant in the French 
army. The modern, corresponding statement, which ap-- 
plies broadly to all forms of cycles, conducting trans- 
formations between any possible pairs of energy-forms, is 
the following : — 

The Third Law of Energetics 

For the continuous, cyclical transformation of any pri- 
mary into any secondary form of energy, within the con- 



THE CYCLE 53 

fines of any limited portion of matter, four things are 
essential: — 

I. A source of supply of the primary form ; 

II. An absorbent of the primary form ; 

III. A positive difference of intensity between the two; 

IV. An absorbent of the secondary form. 

(In the third condition the word " positive " is understood 
to apply when the intensity of II is subtracted from the 
intensity of I.) 

As with the other general laws of energetics, when the 
two forms of energy concerned are heat and work re- 
spectively, this general law becomes one of the special 
laws of thermodynamics, viz. : — 

The Third Law of Thermodynamics 

For the continuous, cyclical transformation of heat into 
work, within the confines of any limited portion of matter, 
four things are essential : — 

I. A source of heat ; 
II. An absorbent of heat, or " refrigerator " ; 

III. A positive difference of temperature between the 

two ; 

IV. An absorbent of work. 

To illustrate from natural facts, taking first the simplest 
and most familiar form of useful cycle, that between the 
potential mechanical energy of water-power stored in a 
mill-pond and the kinetic mechanical energy of moving 
machinery, it is obvious that the four essentials here are : — 

I. The mill-pond ; 

II. The tail-race ; 

III. A difference of level, or "head," between the two; 

IV. The driven water-wheel. 



54 THE THERMODYNAMICS OF HEAT-ENGINES 

Remove from the list any one of the four and it is obvious 
that the whole process must come more or less promptly 
to a stop. 

Similarly, in thermodynamic work, the four essentials to 
the conversion of heat into work are already familiar to the 
student, although their exact identity and relations have 
probably never been pointed out to him. Taking the 
steam-boiler and steam-engine as an example, they are : — 

I. The furnace ; 
II. The "refrigerator," or condenser, which may be 
the atmosphere or may be a current of cold 
water ; 

III. A difference of temperature between the two ; 

IV. The steam-engine piston and the resisting machin- 

ery which is attached to it. 1 

1 It is of further and much wider interest and value to call attention to the 
application of the same law to transformations between other pairs of energy- 
forms from the list given on page 20. The study of electrical engineering 
gives rise to a large number of cycles between electrical energy upon one hand 
and thermal, chemical, or radiant energy on the other, in each of which the four 
essentials may be traced. The study of chemistry, and particularly of organic 
chemistry, is productive of a much greater number of cycles ; for there occur 
a much larger number of chemical compounds of different degrees of affinity 
than the number of different degrees of electrical potential which are com- 
monly in use in the arts. Of all such instances by far the most interesting and 
in one way the most simple, although in other ways it is the most complex, is 
the cycle between animal and vegetable life. The broadest chemical basis upon 
which it rests is that of the oxidation of carbon and hydrogen, with the ob- 
verse dissociation. It also rests broadly upon the chemistry of the nitrates, 
and a myriad of other chemical reactions enter into the complete scheme; but 
none of them could go on without the oxidation which is the basis of the 
whole. 

In this broad cycle it is the office of vegetable life to produce intensity of 
chemical affinity by the dissociation of the carbonic acid of the air into carbon 
and oxygen and of the water of the earth into hydrogen and oxygen, by the aid 
of sun-heat acting through the medium of the chlorophyl of the leaves. The 
oxygen is set free to the atmosphere and the carbon and hydrogen are stored 
in the plant in the form of various hydrocarbon compounds. < 



THE CYCLE 55 

The Reversed Cycle. — If attention be returned to the 
rectangular cycles of Figs, i and 2, it will be clear that 
nothing in the discussion of them has limited their dimen- 
sions. Vertically or horizontally, their limits may be sep- 
arated by as great or as small a distance as accidental 
circumstances may dictate. This being so, let it be imag- 
ined that the vertical sides of the rectangle be mutually 
approached. Carried to the limit this process will result 
in their coincidence. If the displacement of the sides be 
still further continued, it must result in the rising side 
being to the right of the falling side. In order that the 
horizontal processes may close the cycle they will have to 
be reversed in direction ; in other words, the direction of 
rotation of the entire cycle will have altered from a clock- 
wise to a counter-clockwise one, as in Fig. 5. This means 
that at the upper level what was formerly the source of 
energy has become an absorbent ; what was originally the 
absorbent has become a source of supply. 

If computation of the areas measuring the energy-quan- 

It is the office of animal life to produce intensity of mechanical energy and 
of animal heat, to the destruction of the intensity of chemical affinity just re- 
ferred to, by combining within its body the hydrogen and carbon taken in as 
food with the oxygen of the atmosphere. At the same time, carbonic acid is 
rejected into and is stored up by the atmosphere. 

Thus the two forms of life work together, to the destruction by each 
of the form of energy which it consumes and to the production of the 
form which the other consumes. Neither, so far as we know, could exist 
without the other. By the balance between the two the equilibrium of 
earthly conditions is maintained. Energy, intensity, and life are alike 
conserved. 

It is not the purpose of this work to develop farther the study of such cycles. 
It is all-important, however, that the student should see clearly that the laws 
which are under discussion in the present chapter are not narrow ones, apply- 
ing only to certain mechanical devices, but are broad principles which cannot 
be broken nor varied by any accident of form or locality whatever. In this 
broad sense, heat-engines are like all other portions of the universe and all 
forms of life or activity are like heat-engines. 



56 



THE THERMODYNAMICS OF HEAT-ENGINES 



tities be made anew, it quickly becomes evident that the 
energy supplied at the lower level is less than that rejected 
at the upper level by the area of the cycle. According to 
the First Law, that of the Conservation of Energy, the 
difference must be supplied from without. A comparison 
of the energy transformed on the rising side of the cycle 
with that reversely transformed on the falling side shows 

T 




Fig. s 

that the former is now the greater, also by the area of the 
cycle. Plainly, there has been a net transformation of 
energy from what was formerly the secondary form back 
into the primary form. 

Such a cycle is called a reversed cycle, and all of the 
arguments and laws which were developed for the clock- 
wise cycle now apply as rigidly to the reversed cycle if the 
terms be properly interchanged to fit the case. For in- 
stance, referring to the Third Law of Thermodynamics, 



THE CYCLE 57 

page 53, it will be noticed that the word positive therein 
must now be changed to negative. 

In reality, such a cycle should properly be called an 
"obverse" cycle; for it is merely the portrayal of the 
cycle of the secondary form of energy instead of the pri- 
mary. For every "reversed" or counter-clockwise cycle 
of a secondary form of energy which takes place there 
must exist a corresponding " direct " or clockwise cycle on 
the part of a primary form to furnish it motive power. In 
short, — 

A clockwise cycle always portrays the change taking place 
in a primary, a counter-clockwise in a secondary, form of 
energy. 

This is a direct corollary of the Second Law of 
Energetics. 

In thermodynamic work the reversed cycle is that em- 
bodied in the refrigerating or artificial-ice machine. Such 
a machine absorbs heat along the lower temperature-level 
CB, from the beef to be cooled or from the water to be 
frozen ; it lifts this heat up-temperature against its will, 
up the adiabatic BA ; it rejects it at the upper level AD 
into the absorbent circulating- water ; the remnant of un- 
rejected heat at D falls down-temperature to C. This 
cycle requires mechanical power to maintain it in 
motion. Such power is usually derived from a steam- 
engine. 

Efficiency of the Cycle. — It is to be noted, as of especial 
importance, that if the cycle be rectangular, as drawn in 
Figs. 1 and 2, the efficiency of conversion of energy from 
one form to another is given by the ratio between the 
temperature-range within the cycle and the temperature 
of the supply of heat. The energy supplied is meas- 
ured by the area beneath DA, the curve of supply ; the 



58 THE THERMODYNAMICS OF HEAT-ENGINES 

energy transformed is the area of the cycle, or DABC. 

Therefore 

j 7 _ DABC _ fall in intensity utilized , s 

area beneath DA total intensity 



This gives, for weight-engines, 



F = 


H x -H<> 


H x 


F— 


t;~t, 




Ti 



(13) 



and for heat-engines, F= * — 2. (^4) 



This equation is of fundamental importance, and permits 
us to deduce the broad law : — 

In any rectangular cycle of energy-transformation the 
portion of the primary energy supplied which is converted 
into secondary is given by the proportion existing between 
the range of primary intensity during the cycle to the initial 
intensity of supply. 

When the two forms of energy concerned in the rec- 
tangular cycle are heat and work respectively, the above 
principle becomes 

The Fourth Law of Thermodynamics 

In a heat-engine working on the rectangular cycle, the 
heat converted into work is to the heat supplied as the 
cyclical range in temperature is to the original absolute 
temperature of supply. 

The Efficiency of the Reversed Cycle. — Approached 
mathematically, the quantities occurring in a reversed rec- 
tangular cycle similar to Figs. 1 and 2 must bear the same 
relation as they do therein. It is not possible, however, to 
deduce therefrom an expression for its efficiency in the 



THE CYCLE 59 

usual sense, similar to equations (13) and (14), because the 
energy transformed is not now a portion of the energy 
supplied along the lower level. It may obviously bear any 
imaginable relation to the latter, according to conditions of 
relative levels of supply and rejection. If the cycle be 
rectangular the heat absorbed from the body to be chilled, 
or Q%, must bear the same ratio to the energy to be sup- 
plied in lifting this heat to the 7^-level, or Q w , as T 2 bears 
to T x — T v This ratio may obviously be anything 
whatever. 1 

Comparative Inefficiency of Cycles other than Rectangular. 
— Bringing into the argument the law proved on page 51, 
that the heat handled is measured by the area beneath the 
curve representing the heat-process, the Third Law will 
be found not to apply when the cycle is any other than 
rectangular. Thus, let EFGHIKLM, Fig. 6, be any 
rectilinear, but not rectangular, cycle. It may be con- 
sidered as made up of a series of triangular departures 
from the circumscribing rectangle DABC. In such a 
cycle all of the heat must be supplied along the line 
EFGH; for only during this portion of the cycle is the 
entropy increasing. The abstraction of heat must occur 
only along the line IKLM, for here alone is the entropy 
decreasing. Therefore Q v the heat supplied, is measured 

1 Herein appears the fallacy of a frequently recurring error in publications 
concerning artificial ice-making and refrigerating. Thus, it is often stated 
that, since it requires a certain number of B. t. u. to melt a certain quantity 
of ice into water at a given temperature, to make ice again from that same 
water would require, in a perfect refrigerating machine, this same number of 
B. t. u. But between these two quantities of heat: (1) The heat required to 
melt the ice, which when once in, must be lifted out again in order to freeze 
it ; and (2) The energy required to do this lifting, there is no direct con- 
nection. If the heat had to be lifted to a high temperature, as might be the 
case in India, a perfect machine might require much more than this amount of 
heat. If it had not to be lifted far, as would be the case in Canada, a quite 
imperfect machine might require less energy to operate it. 



6o 



THE THERMODYNAMICS OF HEAT-ENGINES 



by the area dEFGHa ; Q 2 , the heat abstracted, is meas- 
ured by the area alKLMd. Hence the efficiency, 

f = Q^-Q<l = EFGHIKLM 



<2 2 



dEFGHa 



Comparing the values for this equation with those for 
the circumscribing rectangular cycle between the same 
limits, DABC, it is plain that Q x has been diminished and 

T 




Fig. 6 

<2 2 increased. The effect of either of these changes alone, 
or of both in combination, is to diminish the value of the 

fraction * ~ , which expresses the efficiency of the 

rectangular cycle. 

It is next important to note that : — 

Between any two temperature-limits the only possible ir- 
regidar clockwise cycle is one lying wholly between those 
limits. 



THE CYCLE 



6l 



Thus, in Fig. 6 it is quite possible to imagine the irregu- 
lar clockwise cycle EFGHIKLM operative between the 
limits T x and T 2 ; for the heat can readily fall into the 
carrier from DF as it passes along EF, and from GA as 
it passes along GH, or out of the carrier into BK as it 
passes along IK, etc. But if we attempt to imagine the 
cycle EFGHIKLM of Fig. 7 operative in a clockwise 

T 




Fig. 7 

direction between the limits T\ and T\ it becomes obvious 
that the thing is impossible. The carrier cannot pass 
along hEFGH absorbing heat from hH\ it cannot pass 
along mIKLM rejecting heat into mM. The Second Law 
forbids. 

The irregular counter-clockwise cycle MLKIHGFE of 
Fig. 7 is quite possible, however, between the limits T\ 
and T\. The carrier, as it passes along HGFE, can easily 



62 THE THERMODYNAMICS OF HEAT-ENGINES 

reject heat to Hh ; as it passes along MLKI it can absorb 
it from Mm. Therefore, 

Between any two temperature-limits the only possible 
counter-clockwise cycle is the rectangular one ; any irregular 
one must vertically circumscribe that rectangle. 

But in such circumscribing cycle the energy required to 
maintain operation is the area of the cycle. The effective 
result is measured by the inscribed rectangle. The effi- 
ciency of the former is plainly less than what it would 
have been had it been rectangular. 

By assuming these triangular departures from the rec- 
tangle to become infinite in number, the proof can be 
extended, with the aid of the calculus, to cover a cycle of 
any non-rectangular form, such as ABDC, Fig. 4. It 
therefore follows, from pages 60, 61, and 62, that — 

Between any two given temperature -limits the rectangular 
cycle is the one of maximum efficiency. 

This proposition may also be proven from general con- 
siderations. If the temperature at which heat enters the 
engine be at any time lower than that of the heat-supply, 
the heat must fall into the engine unrestrictedly. If at 
any time the temperature at which heat leaves the engine 
be above that of the refrigerator, or the source of cold, 
heat must fall out of the engine into the refrigerator. But 
since the sole source of mechanical work lies in the fall of 
heat in temperature, it is obvious that any free or unre- 
stricted fall of the heat down-temperature must represent a 
lost opportunity. The efficiency of the process must there- 
fore be less than when no such free fall takes place. Such 
a lost opportunity is commonly called a " free fall" of heat. 

Conditions for Efficiency. — It is said of water-wheels, and 
is proven by deduction from a line of argument exactly 
parallel to the above, and also by experiment, that the 



THE CYCLE 63 

attainment of the maximum possible efficiency depends 
upon — 

I. The admission of the water at the head-race level 
and without velocity or impact; 
II. The discharge of the water at tail-race level with- 
out velocity; 
III. The handling of the water between the two levels 
without leakage. 

Similarly, it is truly said of heat-engines that the con- 
ditions of maximum possible efficiency are — 

I. The admission of all of the heat isothermally at 
the temperature of supply; 
II. The discharge of all of the heat isothermally at 

the temperature of the refrigerator ; and 
III. The vertical or adiabatic transfer of the heat {i.e. 
without gaining or losing any heat on the way) 
from one level to the other. 

A heat-engine which receives heat from the source of 
supply after having started down the path to the lower 
level (as, for instance, through the reheating coils supply- 
ing live-steam heat to an intermediate receiver) is like a 
water-wheel with a leaky flume which lets water dribble 
into the buckets when they are halfway down to the tail- 
race ; the process is illustrated in GH, Fig. 6. 1 The evil 
of beginning to drop the temperature of the heat before it 
has all safely arrived in the engine, as by wire-drawing, 
belongs to the same class. 

The line IK of Fig. 6 illustrates the loss due to drop- 

*It is an incidental result of the use of reheater coils that another much 
larger loss is modified thereby, and therefore the advisability of their use remains 
an open question; but it is none the less true that the fall in temperature of 
the live-steam heat to the temperature of the receiver must ever be an irreme- 
diable loss, which may or may not be counterbalanced by its beneficial effect 
upon other inevitable free-fall losses incurred at other points in the process. 



64 THE THERMODYNAMICS OF HEAT-ENGINES 

ping the heat out of the engine into the refrigerator before 
it has reached the lower limit by vertical, rectilinear fall, 
as in an engine working with incomplete expansion ; it is 
like pouring the water out of water-wheel buckets before 
they have reached the level of the tail-race. 

The line LM shows the evil of beginning to raise the 
buckets before they are completely emptied ; it has no 
familiar parallel in the steam-engine. 

The line EF shows how work is lost by pouring water 
into the buckets as they are rising ; the corresponding loss 
in the steam-engine occurs in the heating of the feed-water 
in the boiler. 

The rectangular cycle was first defined and its efficiency 
established by Carnot. It is therefore known as the Carnot 
Cycle. Because of its high theoretic efficiency it has been 
much sought after. In nearly all such search the limiting 
portion of the law of its efficiency, the opening words in 
the above statement, " Between any two temperature- 
limits," has been entirely lost sight of. The fact was over- 
looked that higher efficiency might be attained as well or 
better by extending the temperature-limits than by seeking 
the rectangular form within certain prescribed limits. For 
the rectangular Carnot cycle is difficult to reproduce in 
practice. In fact, it has always proven impracticable of 
incorporation into an actual working engine. While the 
task might be possible of solution, yet it is certain that the 
expense which it would involve would always more than 
offset any thermodynamic gain which would ensue. 

Nevertheless, the study of the Carnot cycle is of prime 
importance in heat-engine design ; for of all of the myriad 
of variations which may be played by the individual de- 
signer upon the dozen known heat-engine cycles, the ther- 
modynamic value of each can be safely and accurately 



THE CYCLE 65 

measured only by its approach to or departure from the 
rectangular cycle of Carnot. And while the commercial 
value of a design may be quite another thing from its ther- 
modynamic value, yet the latter must always enter into the 
question as a factor of prime importance whose estimation 
by guess is by no means easy or safe. 



CHAPTER III 

THE THERMAL PROPERTIES OF MATTER 

According to the molecular theory, all matter is made 
up of a vast number of minute particles, or molecules, each 
of which possesses all of the chemical characteristics of 
the substance in question. As to the exact size of the 
individual molecule, there is, of course, much difference of 
opinion ; but in view of their infinitesimal size in proportion 
to any dimensions which the eye can perceive, even with 
the aid of the microscope, the various estimates agree 
more closely than might be expected. One authority 
offers the simile that if a pea should be magnified to the 
diameter of the earth each of its component molecules 
would have attained the size of an orange ; or in other 
words, that the average molecule possesses a diameter of 
about one-billionth of an inch. 

The individual molecules of a body are supposed to be 
separated by distances very great in comparison with their 
diameter, and in gaseous matter these distances of separa- 
tion have been likened to those of the solar system in 
comparison with planets composing it. At any rate, the 
component particles or molecules of a body are anything 
else than rigid, quiescent grains ; they have more or less 
liberty to move actively about, and they are always doing 
so. Balanced in equilibrium between the mutually attrac- 
tive forces, called collectively cohesive, and the separative 
forces due to the momentum of their own motion, like the 
earth between its gravitation toward the sun and its 

66 



THE THERMAL PROPERTIES OF MATTER 67 

centrifugal tendency to fly off its orbit on a tangent, they 
are pictured as being in constant vibration, back and forth, 
at all times and in all conditions of matter. 

Of the form, construction, and attributes of the individual 
molecule little is known, even to the degree of safe con- 
jecture. Since they must be in constant collision during 
their vibrations, and as we can conceive of impact only as 
a lack of elasticity which prevents complete restitution of 
mechanical energy when two bodies meet, we are forced 
to conceive the molecule as being absolutely and perfectly 
elastic ; for if it were not so, some of its kinetic energy, 
when in collision, must be returned as heat instead of as 
motion. But as our only conception of heat is this same 
molecular motion, this is a reductio ad absurdum. 

This does npt necessarily imply, however, that they are 
simple units of matter, — homogeneous spheres, or any- 
thing of that sort. On the opposite, while knowing very 
little as to their absolute construction, we are forced to 
believe that it is very complex. One scientific authority 
makes use of the simile that "compared to a molecule a 
grand piano is simplicity itself." Other authorities con- 
ceive the molecule as being a thing as complex in structure 
and as divisible as an astral nebula. In short, since the 
entire range of astronomical science reveals no limit to the 
bigness of the universe, as compared with human standards, 
there is no basis for imagining any limit to the smallness 
of things. In either direction the infinite and the infinitesi- 
mal stretch into the immeasurable and the unimaginable. 

Each known chemical form of matter is found in at least 
three distinct physical forms : solid, liquid, and gaseous. 

In solids the component molecules are in some way so 
closely bound together, and their vibrations, though still 
taking place, are so narrowly and rigidly limited in relation 



68 THE THERMODYNAMICS OF HEAT-ENGINES 

to one another, that the external form of the aggregation, 
the body taken as a whole, is capable of resisting the action 
of forces of considerable magnitude which may tend to 
deform it. 

In liquids the molecules, though still held together by 
a mutual attraction which limits their distance of separation 
(in so far as internal molecular forces are concerned), they 
are now quite free to assume any position relatively to one 
another which they choose, or to roll about one another. 
This rolling constitutes what we call flow. 

In the gaseous form the molecules are very much more 
widely separated than in liquids or solids, they are more 
actively in vibration, and they tend to separate to the widest 
possible degree permitted by external resistances, such as 
the walls of a containing vessel. 

There is a condition of gaseous matter, when it is in 
contact with the same matter in liquid form, which is 
called vaporous ; and the gradation from a vapor into 
a gas is always more or less gradual. But this does not 
constitute a proper basis for calling the vapor a distinct 
form of matter, although its characteristics may differ 
quite perceptibly from those of a true gas. In fact, all 
of the so-called "permanent" gases are nothing more 
than vapors very highly superheated above their boiling 
points. 

A gas existing under so great an attenuation that each 
molecule travels an appreciable distance without collision 
with another is said to be radiant in form. 

As defined in the tabular list, thermal energy, or heat, is 
the energy inherent in the motion of these individual mole- 
cules, yet integrated for the entire aggregation of molecules, 
so that we perceive it and speak of it as being the heat of 
the entire body. But the motion of each particle is con- 
tinually and repeatedly piling itself up as a momentary 



THE THERMAL PROPERTIES OF MATTER 69 

form of potential energy by being brought to rest against 
molecular resistances, instantly reconverting itself into 
molecular motion, as does the pendulum against gravi- 
tational forces ; yet, since there is always a large majority 
of the molecules in motion (for the rest of each is but 
instantaneous), and since there is another distinct molec- 
ular energy which is purely potential, this vibration of 
the molecules will be called the kinetic form of thermal 
energy. It is the energy of periodic, if not continuous, 
motion, contrasted with the energy of a permanent con- 
dition of separation. 

It has been demonstrated, by a line of experiment and 
argument which cannot be repeated here, that the phe- 
nomenon of temperature is a function of the translational 
velocity of this molecular vibration. Since all kinds of 
matter are solid in form at the lowest temperatures, 
become liquid when heated, and finally, when heated to 
a still higher temperature, become gaseous, and since the 
cohesion of a solid is much greater than that of a liquid, 
and that of a gas the least of all, it is easy to conceive of 
the melting of a solid as a breaking-down of the restricting 
cohesive forces by excessive velocity of vibration, and of 
the vaporization of a liquid as the same phenomenon 
carried to a further degree. The velocity of vibration of 
the molecules of the solid, as it is heated more and more 
nearly to its melting-point, may be imagined as putting a 
greater and greater strain upon the molecular, cohesive 
forces which bind the molecules together in a rigid connec- 
tion, until finally those forces are first balanced and then 
overcome, and the molecules take up that freer activity 
which proclaims the substance a liquid. The vaporization 
of a liquid into a gas presents a similar series of conditions 
of equilibrium. 

The situation is by no means so simple as has just been 



JO THE THERMODYNAMICS OF HEAT-ENGINES 

portrayed, and the student must not expect this explanation 
to clarify all of the puzzling problems of physics. It is 
perfectly certain that the thermal motion of matter must 
include and combine several sorts and degrees of motion, 
in the so-called "different degrees of freedom," involving 
different portions of the mass of matter present in the mole- 
cule. But for the purpose of gaining a clear general con- 
ception of how energy is absorbed by water in its variations 
in temperature and its transformations into ice and steam, 
and of how such energy may be nothing more than mass 
in motion, the above will be found to be a material aid. 

If heat be kinetic energy of molecular translation and 
temperature be a function of the velocity of the same, then 
it is obvious that the temperature of a body cannot be in- 
creased without the addition of energy from without. This 
is true. A body can be warmed by rubbing it, by pound- 
ing it, by compressing it, or by simply heating it. In either 
case the velocity and energy of its vibrations are increased by 
adding more motion from without. Moreover, a definite in- 
crease in velocity of such vibraticn ought to absorb a definite 
amount of energy from without ; and so it does. This is 
so true that heat is accurately measured by means of the 
amount absorbed by some representative substance chosen 
as a standard, when raised through a definite change in 
temperature. The substance chosen is water to the amount 
of one pound avoirdupois, and the rise in temperature is 
defined as that from 59 to 60 degrees Fahrenheit. This 
amount of heat-energy is called the British thermal unit 
(ordinarily abbreviated as " B.t.u.") and is the basis of 
nearly all heat-measurements in English-speaking countries. 

The Mechanical Equivalent of Heat. — As it was stated, 
as the primary law of thermodynamics, that there must 



THE THERMAL PROPERTIES OF MATTER 7 1 

always be an exact equality between the amount of heat 
disappearing and of work appearing, or vice versa, when- 
ever energy changed costume from one to the other, this 
equality must be capable of statement in terms of our 
chosen units of measurements. This is also true. The 
numerical statement is that 

i B.t.u. = 778 foot-pounds. 

This ratio, of 1 to 778, is known as The Mechanical Equiv- 
alent of Heat, and forms the foundation-stone of modern 
thermodynamics. Long suspected by various physicists, it 
remained for the Englishman, Joule, to definitely prove the 
fact and determine the ratio. Joule's methods were crude 
as compared with those of to-day, and he obtained results 
which varied quite widely, as we now judge accuracy. But 
his results were so consistent, after endless repetition, that 
the equivalence was proven beyond a doubt and the law of 
the conservation of energy placed for the first time upon a 
sound foundation. The ratio 1 : 778 is therefore frequently 
referred to as " Joule's equivalent." 1 

Specific Heat and Latent Heat. — If the heat absorbed by 
one pound of water during a rise of one degree Fahrenheit 
be measured at different temperatures, it will be found to 
be not always quite the same. Yet it remains very close 
to unity, varying not one per cent between 6o° F. and the 
boiling-point, and by barely five per cent through the entire 
range of temperatures common to steam-engine work. Its 
value at any temperature may be found by consulting the 
first column of the steam-table. 

If one pound of ice at a temperature of 31 F. be heated, 
with the object of reducing it to water at 33 F., it will de- 
velop that it absorbs very much more heat than 2 B.t.u. 
Indeed, it will be apparent that it absorbs as much heat in 

1 Grindley (Trans. Phil. Soc., Vol. 194) prefers the value 774. 



72 THE THERMODYNAMICS OF HEAT-ENGINES 

this slight change of temperature as it does in afterwards 
being heated up to a temperature of 175 F., or through a 
rise of 144 degrees. The explanation of the apparent in- 
consistency lies in the fact that in the first case the water 
has changed its physical state from ice to water. In fact, 
if the ice be merely melted, from ice at a temperature 
of 32 to water at a temperature of 32 , without any rise 
in temperature whatever, it will be found that it absorbs 
143 B.t.u. 

This quantity of heat is known as the latent heat of fusion. 
It is apparently absorbed in a separation of the molecules 
against the cohesive forces which bind them together, the 
kinetic energy added from without being converted into the 
stored motion of separation. The process cannot consist 
of a change of velocity of molecular vibration, for there is 
no change in temperature involved; the energy absorbed 
is not revealed by the thermometer. It consists of the at- 
tainment of a mean position of greater separation of each 
molecule, relatively to rest, which has been attained by 
motion against resistance, by separation against cohesion, 
and is therefore purely potential in its character. It is, in 
fact, the potential form of thermal energy. 

This sort of heat the thermometer does not perceive. 
Neither do our nerves. We can feel that one body is 
hotter or colder than another, or hotter or colder than our 
fingers. But if ice were not harder to the touch than water, 
the human perceptions would not reveal the difference 
between ice at 32 F. and water at the same temperature. 
Yet the latter possesses 143 B.t.u. more. than the former. 
The energy involved in a change of temperature in a sub- 
stance is therefore known as its "temperature-heat" or 
" sensible heat" or "thermometer heat," while that involved 
in a change in its physical structure is known as its " latent 
heat " (of fusion or of vaporization as the case may be). 



THE THERMAL PROPERTIES OF MATTER 73 

The first is kinetic in character ; the second is potential. 
The first is perceptible by our senses ; the second is entirely- 
hidden from them. 

So far reference has been made only to water. But the 
same description applies to nearly all forms of matter, ex- 
cept that the quantities of heat involved are quite different. 
Thus, if 100 pounds of cast-iron be put into a tub containing 
100 pounds of water, the iron having a temperature of 140 F. 
and the water being at 50 , it will be found, when the two 
have come to the same temperature, that they are not at 
95 , as might be expected, but at 6o°. In other words, 
instead of the iron having fallen 45 in temperature while 
heating the water 45 , as would seem natural, the iron has 
fallen 8o°, while the heat which it has lost has sufficed to 
raise the water only io°. This shows that iron absorbs or 
gives out, during a given change of temperature, only about 
one-eighth as much heat as does water. 

If the experiment were repeated with other substances 
than iron, a different result would be obtained in each case, 
showing that the amount of heat absorbed per degree 
change in temperature is different with each substance con- 
sidered. This amount of heat, measured in B.t.u., is called 
the specific heat of the substance. A tabular list of the spe- 
cific heats of various substances used in steam-engineering 
will be found in the Appendix. 

If the amount of heat absorbed in the melting of one 
pound of iron, or the vaporization of one pound of alcohol, 
be similarly measured, the results will be different than 
with the melting of one pound of ice or the vaporization 
of one pound of water. It is therefore said that different 
substances possess different latent heats of fusion or vapor- 
ization, as the case may be, as well as different specific 
heats. A tabular list of these heat-quantities will also be 
found in the Appendix. 



74 the thermodynamics of heat-engines 

The Formation of Steam under Constant Pressure 

Pressure and Temperature. — If a pound of water be 
heated in an open vessel, it is found that the temperature 
rises continuously, as heat is absorbed, without perceptible 
alteration in its physical condition, until a temperature of 
about 212° F. is attained. At this point, for some invisible 
reason, the temperature refuses to rise further. Heat may 
be added indefinitely, yet the water becomes no hotter. 
Some result is visible, however, in the formation of vapor, 
and if the heating be continued long enough, it will be 
found that the water has quite disappeared, having been 
completely vaporized. Simultaneously there has occurred 
the absorption of a very large amount of heat. 

Pressure and Temperature. — If, however, the water be 
placed in a closed vessel and a pressure of, say, 100 pounds 
per square inch be applied to its surface by means of an 
air-pump, it will be found that when the water reaches a 
temperature of 21 2° it does not vaporize. Instead, the 
temperature will continue to rise to a point between 337° 
and 3-3 8° before vaporization sets in. There is obviously 
some connection between the pressure upon the surface of 
the water and the temperature at which it will vaporize, 
and this connection is found to be a very rigid and exact 
one. The explanation is that when the molecules attain a 
velocity (or, in other words, a temperature) at which they 
would otherwise break their molecular bonds and take on 
a wider range of vibration, they find themselves opposed 
by other resistance than merely that of molecular cohesion ; 
they are pressed together by the external pressure upon 
the entire mass. They therefore continue to vibrate within 
their prescribed limits until they attain such a higher ve- 
locity that they are able to overcome, not only the molecular 
forces, but the external ones as well. 



THE THERMAL PROPERTIES OF MATTER 75 

The relation between the external pressure and the tem- 
perature at which boiling takes place is not a simple one, 
although it is rigid and exact. It cannot be expressed 
accurately by any mathematical equation. For the sake 
of both accuracy and convenience it is customary to refer 
to the columns of a steam-table for its determination. The 
data found in the steam-table have been derived from 
experiments many times repeated. 

Volume and Temperature. — If account be taken of the 
volume of steam produced during the evaporation of the 
water, it will be found in each case that a definite volume 
has always been developed by the time that one pound of 
water has been just entirely evaporated. This volume is 
called the specific volume of saturated steam. It, too, will 
be found to have different values under different condi- 
tions as to pressure and temperature ; but under the same 
conditions it is always the same. This relation between 
pressure and specific volume, or between temperature and 
specific volume, is also impossible of exact mathematical 
statement, though in itself rigid and exact. It is therefore 
customary, for the sake of accuracy as well as convenience, 
to refer to a steam-table for each determination. 

Superheat and Saturation. — If the heating of the one 
pound of water in the closed vessel be continued after all 
of the water is evaporated, it will be found that the tem^ 
perature again begins to rise ; and this time it will continue 
to rise as long as heat be added to it. Just at the point 
where evaporation is complete and ' the final rise in tem- 
perature begins, the steam is known as dry saturated steam. 
At any temperature above that it is known as superheated 
steam. At any point between the beginning of boiling and 
complete saturation, when the original one pound is partly 
water and partly steam, the steam is known as wet saturated 
steam. In other words, steam in contact with water is 



76 THE THERMODYNAMICS OF HEAT-ENGINES 

always saturated steam and must always have a definite 
temperature and a definite volume when under a definite 
pressure. Steam separated from the water from which it 
was derived may have any temperature whatever at or 
above the boiling-point, and any volume whatever equal to 
or greater than the specific volume. 

If heat be added to saturated steam, it will become super- 
heated ; if heat be abstracted from it, it will condense. If 
pressure be released from wet steam, more steam will be 
formed ; if the pressure upon it be increased, some will 
condense. 

Suppose that the one pound of water and steam should 
originally consist of y pounds of steam and of I —y pounds 
of water, and that the pressure be maintained constant; if 
heat be added y will increase and the volume will increase 
in proportion, but there will be no change in temperature ; 
if heat be abstracted, y will decrease and the volume will 
decrease in proportion, but there will be no change in tem- 
perature. It is therefore obvious that so long as the pressure 
remains constant the volume can be increased only by the 
addition of heat, vaporizing some of the water ; the volume 
can be decreased only by the abstraction of heat, condensing 
some of the steam. 

If the volume be maintained constant, the pressure can be 
increased only by the addition of heat, which is absorbed 
in two ways: (i) By the rise in temperature of the entire 
mass ; (2) By the vaporization of a portion of the water, 
which fills the fixed volume with steam of greater density 
and pressure. If the volume be maintained constant, the 
pressure can be decreased only by the abstraction of heat, 
with the consequent cooling of the entire mass and the 
condensation of some of the steam. 

In these phenomena the student will note a distinct 
difference of action between these steam-and-water mix- 



THE THERMAL PROPERTIES OF MATTER JJ 

tures and the permanent gases which he has probably- 
studied in his physics. 

The total amount of thermal energy which is taken up 
by one pound of water in changing from water at 3 2° F. 
to dry saturated steam at any higher temperature obviously 
consists, according to the foregoing argument, of three 
portions, which are entirely distinct in their origin and 
characteristics, viz. : — 

I. The energy absorbed in increasing the temperature of 
the mass, or the velocity of vibration of the molecules. 
It is kinetic, not potential, in its character ; it is purely 
molecular, or internal, i.e. it is dependent upon merely the 
relative motion of one molecule to another, and is entirely 
independent of any circumstances or conditions external to 
the body. It is called " vibration-heat" or " temperature- 
heat" or "sensible heat" or "heat of the liquid." It is 
represented by the algebraic character (/), and is presented 
in the steam-table in the second column. 

II. The energy absorbed during vaporization in separating 
the molecules one from another against molecular forces. It 
is potential, not kinetic, in character. It is purely molecu- 
lar, or internal. It is called " disgregation work" the term 
coming from the same Latin origin as the word congre- 
gation ; the latter signifies the gathering together of a 
flock, the former the scattering of a flock (of molecules). 
It is given in the steam-table in the third column, and is 
represented by the letter D. 

III. The energy absorbed in separating the molecules, 
one from another, against the resistance due to any external 
pressure which may be forcing them together. It is poten- 
tial, not kinetic, in its character. It is largely external, 
i.e. it depends upon the amount of this external force and 
the distance to which it is forced back. Thus, in the 
ordinary vaporization of water under atmospheric pressure 



?8 THE THERMODYNAMICS OF HEAT-ENGINES 

it consists of the separation of a certain volume of air from 
the earth against gravitation, and depends in amount upon 
the atmospheric pressure prevailing at the given time and 
place. It is called the " external work." It is represented 
by the letter X, and is presented in the steam-table in the 
fourth column. 

Internal Energy. — Upon examination it is to be seen 
that the first two of these three quantities of energy possess 
a distinct characteristic in common, viz. : they are both 
" purely molecular, or internal." They are therefore lumped 
together into a single heat-quantity which is called the 
" internal energy " or the " internal work " of the body. 
It is given the algebraic symbol /. 

It is obvious, from the definitions just given, that the 
amount of this energy is independent of external circum- 
stances. It depends solely upon the temperature, the 
physical state, and the peculiarities of the body in question. 
Thus, with steam, if there be present a certain quantity of 
steam at a given temperature and pressure, the amount of 
its internal energy can be known without any reference to 
the manner or the external circumstances of the steam's 
production, whether under constant pressure or under any 
other imaginable set of conditions. The total amount of 
heat-energy in the steam, or the amount needed to produce 
it from water of any given temperature, will vary with 
these external circumstances ; but the amount of this heat- 
energy which is properly to be classed as internal energy 
would be the same in any case. The mathematical defini- 
tion of this internal energy is given by the equation 

1=1+ D. 

The Latent Heat of Vaporization. — It is further obvious 
that the last two of the three quantities of energy also 
possess a distinct characteristic in common, viz. : they are 



THE THERMAL PROPERTIES OF MATTER 79 

both potential in form. They are therefore lumped together 
to constitute the potential portion of the heat-energy of the 
body under the name "latent heat," or "latent heat of 
vaporization." Its mathematical definition is given by 
the equation , n 

where L is the latent heat of vaporization. Its values are 
presented in the steam-table in the fifth column. 

Total Heat. — Finally, the "total heat" of the steam- 
mass, or the quantity of heat required to produce it from 
water at 3 2° F., is given by the equation 

H= l+D + X=l+L = I+X. 

The values of H will be found in the sixth column of the 
steam-table. 

The relation between these several heat-quantities and 
the characteristics which connect them may best be seen 
by the following tabular exhibit : — 

Total Heat, or 
Total Thermal Energy. 

Kinetic Energy. Potential Energy, or 

Latent Heat of Vaporization. 



(I) Heat of the liquid (II) Disgregation work (III) External work 

= temperature-heat = potential molecular = potential energy 

= kinetic molecular energy of separation of molecular sepa- 

energy. against molecular ration against exter- 

forces. nal forces. 



Molecular Energy, or 
Internal Work. External Work. 

Total Thermal Energy. 

Having pointed out that vaporization, or its reversed 
process, condensation, takes place normally under fixed 
temperature and pressure, it is only necessary to add that 



80 THE THERMODYNAMICS OF HEAT-ENGINES 

it need not always be present in full amount. The mathe- 
matical definitions given above apply to one pound of dry 
saturated steam ; but of each pound of water on hand it 
often happens in practice that only a portion has been 
vaporized. The result is what is called wet saturated 
steam, or a steam-and-water mixture. The portion of the 
pound which is water, still unevaporated, is usually referred 
to as " percentage of moisture," and the remainder the 
" percentage of dryness." As the vaporization proceeds, 
from ofo of steam to loofo of steam, the water appears to 
take up the heat and become vapor, molecule by molecule ; 
and each particle of water of equal weight, in becoming 
vapor, takes up an equal amount of heat and produces an 
equal volume of steam. It therefore follows that if the 
latent heat of vaporization of one pound of water be repre- 
sented by L, and the volume produced by its complete 
vaporization be V, then, if only the fraction y of the pound 
be already evaporated, leaving i — y water, the latent heat 
of the steam-and-water mixture must be yL, and its volume 
y V. This statement holds true if the original volume of 
the pound of water be considered zero, and it is so small 
in comparison with the steam-volumes formed from it that 
this course seldom introduces a perceptible error. 

Finally, if we represent all of the heat in the steam, or, 
more strictly speaking, all the heat required to convert it 
from water of a temperature of 3 2° F. into wet saturation 
at the given temperature or pressure, by H' , and call it the 
" total heat " of the steam-and-water mixture, we have for 
wet steam the equation 

H> = 1+yL = 1+yD+yX. 

The Absolute Zero. — In all of these steam-table heat- 
quantities it must be remembered that an entirely arbitrary 
zero of heat and temperature has been assumed, viz. : the 



THE THERMAL PROPERTIES OF MATTER 8 1 

melting-point of ice. But water at 3 2° F. does not contain 
zero heat. It contains a great deal of heat, as will quickly 
appear if one attempts to remove the heat in order to con- 
vert the water into ice. As noted above, 143 B.t.u. can 
be removed before it is frozen, and the ice is then no colder 
than 32 F. Moreover, this ice still has considerable heat, 
for this heat can be drawn out of the ice and proves itself 
to be heat. This process may be continued until the ice 
is at the temperature of o° F. But even here it is obvious 
that more heat can still be abstracted from the ice ; so that 
the Fahrenheit zero is plainly no real zero at all, but only 
a starting-point from which measurements can easily be 
made. 

So the process of heat-abstraction may be continued, 
while the ice continually falls in temperature as it gives up 
its heat, until it has reached a temperature about 461 ° be- 
low the Fahrenheit zero. No one has ever cooled ice, or 
anything else, to this point ; but it is known, by lines of 
physical experiment and argument which need not be pro- 
duced here, that when this point were reached it would be 
found that the ice actually contained no more heat to be 
abstracted, its molecules would actually be at complete 
rest. This point is called the " Absolute Zero " of temper- 
ature, and the total heat of a body, in its exact sense, is 
always the amount of heat required to bring it from a tem- 
perature of absolute zero to its given condition. But for 
purposes of convenience it is much better to measure all 
heat-quantities as a surplus above 32 F. So the steam- 
tables are built on this plan. If the fact of the absolute zero 
be kept clearly in mind, there is no objection to be found. 

The Isomorphic Curve. — The mathematical definition of 
entropy was given (page 39) as 

dN = ^$. (2) 



82 THE THERMODYNAMICS OF HEAT-ENGINES 

Similarly, the mathematical definition of specific heat is 

s = T^rr> or Q = S ( T '- T ")> 05) 

where 5 is the specific heat of the body in question and Q 
the quantity of the heat involved in altering its temperature 
from T' to T" . When the alteration in temperature of 
the body and the heat concerned therein become less than 
any assignable quantity, this equation becomes 

dQ = SdT. (16) 

Equating equations (2) and (16) gives 

dQ = TdN= SdT, 

dN=sf. (17) 

Integrating, 

W=Sj^=S\og e T, (18) 

if 5 be a constant, as it is very approximately with most 
substances. Taking limits, this becomes 

N- N = S(\og e T - log, T Q )= Slo ge ^. (19) 

This is the general fundamental equation between en- 
tropy and temperature, during changes : — 

(1) When the specific heat of the body remains constant, 

and 

(2) When the body undergoes no alteration in physical 
state, all of the heat being absorbed in alteration of tem- 
perature. 

Should the physical state be altered by addition of heat, 
the process is isothermal, T becomes a constant, and the 
integration of Equation 2 gives 

N=Q. (20) 



THE THERMAL PROPERTIES OF MATTER 



83 



If Equation 19 be plotted on an entropy-temperature 
diagram, as in Fig. 8, taking A as representing the initial 
condition of the heat, it must reveal a curve rising toward 
the right, as along A W; for if the body be heated without 
change in physical state, increase in temperature must 
mean addition of heat, and that again must mean increase 
of entropy. This curve A W might then be called the 
" curve of heating and cooling " and is frequently referred 



T 


< 


lo 


w 

/ 
/ 

bL- - 

i\ 

/! 
/ J 


s jo 

ycy; 

1 I 1 \ 






• )■ 


A 1 


I ,1 v - 
' s 

! ' 1 






/! 




" 1 1 




, 


/! 




1 " 






/ ! 




1 i' 
1! 






/ 1 




i| 






/ 1 




1 ' 1 






/ ! 




! '! 






S 




1 i' 




k — 


l\ 


a \b 


y\ cl'd. 







Oo 




N 



s 



Fig. 8 

to by that name; but as it is the curve on the diagram 
which represents heat-changes in a body only without 
alteration in physical state (that is, from solid to liquid, 
liquid to gaseous, etc.), a preferable name is the isomorphic 
line, by which name it will be referred to hereafter in this 
work. Its equation is always of the form of Equation 19 : — 

1 
in which the zero-subscripts refer to the original condition 

of the body. 



84 THE THERMODYNAMICS OF HEAT-ENGINES 

If the initial temperature of the heat-change be assumed 
to be zero, the value of N from Equation 19 becomes 
infinity for any finite final temperature whatever. The 
conclusions forced upon us by this fact are : — 

(1) That all absolute values of entropy for ordinary tem- 
peratures are very large indeed in comparison with the 
changes in entropy-quantity accompanying all ordinary 
temperature-changes ; and 

(2) That Equation 19, which is based upon the supposi- 
tion of constant specific heat, cannot hold true for those 
regions of temperature adjacent to the absolute zero which 
have not yet been explored as to specific heats, etc. 

If the point O, Fig. 8, represent the heat-condition of one 
pound of ice at the temperature of absolute zero, — that is, 
having no heat, no temperature, and no entropy, — and if 
heat be imparted to it, the process will be represented by 
some such line as Okl. The left-hand portion of the curve 
is indeterminate ; but without question the portion near to 
the temperature of 3 2° F. follows Equation 19, the value 
of 6" being the specific heat of ice. Having attained this 
temperature, at the point I, the further addition of heat no 
longer increases the temperature; instead, the ice melts at 
constant temperature, absorbing much heat in the form of 
latent heat. This isothermal process is represented by the 
straight horizontal line I A. 

At the point A, therefore, the diagram represents the 
thermal conditions of melted ice at 32 F. This, for con- 
siderations purely of convenience, has been chosen as the 
arbitrary zero of heat for the construction of the steam- 
tables. Water at this temperature, or 492. 8° F. absolute, is 
arbitrarily assumed to have no heat. In reality it has much. 

Equations 17 to 20 everywhere make use of the tem- 
perature-factor as an absolute quantity and never as a 
relative one. The entropy-factor, on the contrary, always 



THE THERMAL PROPERTIES OF MATTER 85 

appears as a relative quantity, its absolute value being 
indeterminate. It is therefore quite allowable to measure 
all values of N from a purely arbitrary zero, such as that 
show by the axis O T through the point A representing 
melted ice. The zero-axis of temperature, however, must 
always appear in its true, absolute position, as at O N. 

If further addition of heat to the melted ice be made, the 
process must again follow the isomorphic A W, and will 
continue to do so until the temperature is able to overcome 
the resisting pressure and cause evaporation. Should this 
point be reached at B, the further addition of heat will 
result in the isothermal evaporation of the water, under 
constant pressure and temperature, along the line BC to C, 
at which point it has all been evaporated into dry saturated 
steam. Further addition of heat again causes rise in tem- 
perature along the isomorph CD of superheated steam. 
This curve will continue unmodified until the temperature 
of dissociation is reached, — a point outside the province 
of the present discussion. 

Each of the curves KI, AB, and CD has the same equa- 
tion, viz. : Equation 19 ; but in each the value of 5 is dif- 
ferent, being in the several cases the specific heats of ice, 
water, and steam respectively. 

The temperature-level at which the evaporation BC 
occurs depends upon the pressure upon the surface of 
the water. As there may be an infinite number of such 
pressures, differentiated by infinitesimal steps, so there 
may be an infinite number of horizontal isotherms such 
as BC, separated vertically by infinitesimal temperature- 
steps. The infinite number of C-points found in such a 
series would all lie in a locus such as 55. Each point in 
this locus represents the thermal condition of one pound 
of saturated steam of the given temperature. It is there- 
fore known as the Line of Saturation. 



86 THE THERMODYNAMICS OF HEAT-ENGINES 

The adiabatics, as Bb, Cc, the isotherms, as I A, BC, and 
the isomorphs, as AB, CD, all represent natural processes 
frequently met with in practice. The saturation-curve 
does not represent a natural process. The alteration 
of steam up-and-down temperature while remaining in a 
condition of dry saturation does not occur in nature, and 
could be artificially enforced only by the most delicate and 
unstable adjustment of heat-supply or heat-abstraction dur- 
ing the process. The locus of saturation-points is, how- 
ever, a great convenience in geometric calculations. 

It is obvious that any point to the right of 5vS must 
represent steam having both more entropy and more tem- 
perature than saturated steam of the same pressure; it 
therefore must represent superheated steam. Any point 
between SS and AB must represent a steam-and-water 
mixture ; for it has more entropy than water and less 
than dry steam. Any point to the left of klAE must be 
meaningless (except in the strictly mathematical sense 
prescribed for all negative quantities); for it has less 
entropy than water or ice of the same temperature, which 
is impossible. 

To this statement must be noted the unimportant excep- 
tion that the exact temperature-level of the isotherm I A is 
somewhat variable, according to the pressure under which 
the melting or freezing takes place. But as a wide varia- 
tion in pressure produces only a slight variation in temper- 
ature, only a very slight area above I A and to the right of 
kl produced suffices to cover all of the territory where a 
geometrical point might represent a physical actuality. 

In the passage of the pound of steam-and-water mixture 
across the isothermal path BC, from all water at B to all 
dry steam at C, the vaporization takes place particle by par- 
ticle, each molecule of water absorbing an equal amount of 
heat, an equal amount of entropy, and producing an equal 



THE THERMAL PROPERTIES OF MATTER 87 

increment of volume. This fact, stated mathematically, 
gives the equation 

1/ = - — = — = — 1 (21) 

where y is the fractional weight of steam vaporized, yL 
the latent heat absorbed therein, n the increase in entropy 
due to the vaporization of w, and v the volume of steam 
produced. The capital letters represent the specific quan- 
tities of the same things applying to the unit of weight. 
Thus, if the point Y, Fig. 8, represent the heat-condition 
of a pound of steam-and-water mixture, then the weight of 
steam present, the heat and the entropy absorbed in its 
vaporization, and the volume produced can all be known 

BY 
by equating the proportion — — to Equation 2 1 ; for in the 

BC 

latter all of the denominators can be found from the steam- 
tables. 

If attention be returned to the tabulations of energy- 
quantities given on pages 77-79, it will be plain that many 
of them must be visible upon Fig. 8 in the graphical 
form of areas. Since the heat handled in any process is 
measurable by the area beneath the curve depicting the 
process and betweeen ordinates (see page 50), the heat 
absorbed in raising one pound of ice from absolute zero to 
the melting-point must be measured by the area beneath 
the curve Okl. That absorbed in the isothermal melting 
of the ice must be measured by the area of the rectangle 
UAa beneath I A. 

This brings consideration to the zero of the steam-tables, 
at A ; for all ice-phenomena lie below its lower limit. 
Starting from melted ice, the heat absorbed in the raising 
of the one pound of water from 32 to any known tempera- 
ture, as at B, which is the heat of the liquid at that point, 
or the heat-quantity /, must be measured by the area aABb. 



88 THE THERMODYNAMICS OF HEAT-ENGINES 

The latent heat absorbed in the isothermal vaporization 
of the one pound of water along BC must be measured by 
the area of the rectangle bBCc. There are, of course, an 
unlimited number of such rectangles between AW and SS, 
according to the temperature-level of BC. It is further 
obvious that Equation 21, applying to any given steam-and- 
water mixture, such as that at F, shows that the amount 
of latent heat present in the mixture is measured by the 
area bB Yy. This also follows from Equation 3 and from 
the argument on pages 50-51. 

The heat absorbed in superheating the steam above the 
temperature of saturation, as to D, must be measured by 
the area cCDd. 

The total heat, as given by the steam-tables, from water 
at the temperature of melting ice to either hot water, as at 
B, to partial evaporation, as at Y, to dry saturation, as 
at C, or to superheat, as at D, must be measured by the 
area between a and the point in question and beneath the 
curve A BCD. 

In all of the foregoing illustrations and arguments it has 
been assumed that the substance undergoing thermal altera- 
tion was what is known chemically as H 2 0. It is obvious 
that the diagram of Fig. 8 and the equations would apply 
equally correctly to any other substance which is known 
to pass successively through the stages of solid, liquid, and 
vaporous in a manner similar to water, provided the proper 
specific heats and other constants be substituted. Such 
diagrams can be applied usefully in the study of the action 
of ammonia, carbonic acid, ether, etc., in the refrigerating- 
machines, and of liquefied air, oxygen, nitrogen, hydrogen, 
etc., in the investigation of low temperatures. For the 
numerical values of the former series reference may be 
had to Peabody's "Steam Tables." Of the latter series 
very little is known empirically ; nevertheless, the diagram 



THE THERMAL PROPERTIES OF MATTER 89 

is exceedingly valuable in aiding the comprehension of the 
rather obscure phenomena occurring in the liquefaction of 
the so-called permanent gases. 

PROBLEMS IN THE USE OF THE STEAM-TABLES 

1. If a correct steam-gage reveal a pressure of 93.7 lbs. per 
square inch and the barometer stand at 29.37", what is the ab- 
solute pressure within the gage? (One cubic inch of mercury 
weighs 0.491 lb.) Ans. 108.12 lbs. 

2. What is the temperature of saturated steam of an absolute 
pressure of 172.3 lbs. per square inch? Ans. 369. 15 . 

3. If the temperature of a body of saturated steam be 148 F. 
and the barometer stand as in Problem 1, what would be a correct 
reading for a vacuum-gage connected with the same ? Ans. 22.19". 

4. What is the " total heat " of the steam of Problem 2 ? 
Ans. 1193.93 B.t.u. 

5. What would be the total heat per pound of this steam were 
it known to be 27% wet? Ans. 963.67 B.t.u. 

6. How, much heat does it require to alter one pound of water 
from a temperature of 62 F. to 187 F.? Ans. 125.5 B.t.u. 

7. How much more heat is required to convert it into steam of 
a pressure of 84 lbs. absolute but 5% wet? Ans. 977.5 B.t.u. 

8. How much more heat (after the process of Problem 7 is 
finished) is required to convert it into steam of the same pressure 
but having a temperature of 468 F. ? Ans. 128.74 B.t.u. 

9. How much external work is done by the heat added in 
Problem 6? Ans. Practically zero. It is so small a quantity 
that it may always be neglected in ordinary engineering calcula- 
tions. If the water were heated under atmospheric pressure, it 
would in this case amount to about 1.07 foot-pounds; under 
other pressures it would be proportional thereto. 

10. How much external work is done by the heat added in 
Problem 7? Ans. 59,320 foot-pounds. 

11. What would be the final volume in the case of Problem 7? 

Ans. 4.927 cu. ft. 



CHAPTER IV 

THE STEAM-ENGINE CYCLE 

Let Figs. 9 and 10 represent rectangular coordinate 
fields measuring temperature and entropy in the first case, 
and pressure and volume in the second case respectively. 
Let us trace on these fields the action occurring in an 
ideally perfect steam-engine. 

It is first to be noted that consideration of mere locality 
of occurrence has nothing to do with thermodynamic phe- 
nomena. Thus, in the actual steam-boiler and engine the 
pressure is first supplied by a feed-pump, then the heat is 
added in the boiler, and finally the steam is removed to the 
engine-cylinder for further thermal action. In following 
the thermodynamics of these processes, however, it is much 
easier if they all be imagined as occurring within the 
engine-cylinder ; that is, that there is first placed therein 
a pound of cold water to which heat is applied, first warm- 
ing it and developing pressure, and finally engendering 
steam and volume. Then let it be imagined that the source 
of heat be removed while the piston takes its stroke ; after 
which condensation is carried on by the application of 
cold within the cylinder instead of within a separate 
condenser. 

In order that our diagrams may agree with the steam- 
tables, it is necessary to confine the attention to the heat- 
phenomena occurring in a single pound of water, and to 
assume that the initial condition of this water coincides 
with the arbitrary zero of the steam-tables, the temperature 

90 



THE STEAM-ENGINE CYCLE 9 1 

of melting ice, or 3 2° F. At this temperature, or 492. 8° 
absolute, the water is supposed to have zero heat, and 
therefore zero entropy. Its volume is 0.017 cu - ft.; but 
this is so small in comparison with the other volumes han- 
dled that it will not appear upon the diagram, and may be 
omitted from all calculations without appreciable error in 
engineering work. At this temperature, too, its pressure 
is too small for exhibition upon the diagram. Its condi- 
tion is therefore properly represented by the point A in 
Fig. 9, and very closely by the point O in Fig. 10. 

As heat is added, both temperature and entropy must 
increase along the isomorphic curve A W. The pressure 
will also increase markedly. The volume will increase 
slightly, but only by a very small percentage of the original 
volume, which is itself too small for visibility on the press- 
ure-volume diagram. The process is therefore properly 
represented by the line OC of Fig. 10, or, with the greatest 
accuracy, by a line parallel and very close to it. 

The point C represents, then, the thermodynamic condi- 
tion of one pound of heated water under a retaining press- 
ure which is equal to its own vapor-tension ; this pressure 
is therefore sufficient to restrain the water from vaporiza- 
tion. In order to study the different processes which may 
develop from this point the following series of problems 
is presented. 

The Cycle of the Boiler-explosion. — Let the point C rep- 
resent the condition of any single pound in a ton of hot 
water contained in a steam-boiler, or in any other similar 
vessel. For the time it is permissible to neglect the pres- 
ence of any steam above the surface of the water, if there 
be any. Let it be supposed that a rupture of the boiler- 
walls suddenly releases from the water the pressure which 
has been upon it. The decreased pressure permits the 
kinetic energy of the vibrating molecules of water to 



9 2 



THE THERMODYNAMICS OF HEAT-ENGINES 



expend their temperature in mutual separation. Vaporiza- 
tion and the performance of external work ensue. 

Because of the instantaneous character of the process 
there is no time for heat-interchange with surrounding 
bodies. The process therefore takes place naturally and 
actually as the pure adiabatic which it tends to be. Such 




Fig. 9 



an explosion is, in fact, one of the few actual instances of 
a purely adiabatic process known in practice. 

Adiabatic fall from C, Fig. 9, develops the plumb-line 
CH. When H is reached, the kinetic energy of the mole- 
cules is in equilibrium with the restraining pressure, which 
is now the atmospheric pressure, and further fall becomes 
impossible. 

The fact that H lies between the water-curve BC and 



THE STEAM-ENGINE CYCLE 93 

the saturation-curve .S-S shows that the one pound of H 2 
is now a mixture of water and steam. The proportion 
of steam (by weight) present is given by the ratio of 
BH -i- Be. The heat still present in the mixture must be 
that beneath the curve ABH. In other words, the heat 
which must be rejected if the water is to be returned to 
its original condition at A must be the area beneath the 
curve HBA. The heat which has been converted into 






Fig. 10 

work must be measured by the area between ABC and 
HBA. This area is the triangle BCH. 

Problem 12. — If a boiler containing one ton of hot 
water under a gage-pressure of 100 pounds per square inch 
should explode, how much energy (in foot-pounds) would 
be released, and what volume of steam would be formed ? 

Let it be assumed that the barometric pressure is 29.50 
inches of mercury. (In actual steam-work the barometer- 
reading must always be known.) This locates B at the 
level of 21 1. 3 F., or 672. i° absolute. The pressure at C, 
being 100 pounds per square inch higher than this, is located 



94 THE THERMODYNAMICS OF HEAT-ENGINES 

by the aid of the steam-table at the absolute temperature 
of 798. 24°. The entropy of B is 0.3116. That of C is 
0.4860. Therefore the entropy BH is 0.1744. 

The heat beneath the isomorphic BC must be the heat 
of the liquid at C minus that at B, or 308.0— 180. 1 = 
127.9 B.t.u. The heat beneath the isothermal HB must 
be the width of this rectangle bBHc times its height, or 
0.1744 times 672.1, which is 117.2 B.t.u. The area of 
the triangle is therefore the difference between these two 
heats, or 127.9 — ll 7- 2 > or IO -7 B.t.u. 

This is the energy released per pound expressed in heat- 
units. The total energy released, expressed in foot-pounds, 
must be — 

10.7 X 2000 x y/8 = 16,649,200 foot-pounds. 

This is about equal to that developed by the firing of a 
modern 6-inch rifle. 

The question as to the volume of steam developed calls 
for reference to Equation 21, page 8y. In applying this 
equation the entropy of vaporization, Be, at the BH\eve\, 
appears from the steam-table to be 1 .4372. The entropy BH 
is 0.1744. Therefore the proportion of steam present is 

' 44 = o. 1213 of a pound per pound. The specific vol- 
ume of vaporization at this same level is 26.98. Therefore 
the total volume of steam formed is — 

0.1213 x 26.98 x 2000 = 6547 cubic feet. 

This explains why boiler-explosions are so completely 
destructive to the buildings enclosing them. The building 
explodes, as well as the boiler, from the immense volume 
of steam formed. The action is so instantaneous that no 
weakest point has time to relieve the rest of the structure 
by giving way first. Each brick is torn from its neighbor, 
and the building collapses into a heap of rubbish. 



THE STEAM-ENGINE CYCLE 95 

The Cycle of the Perfect Steam-engine. — Returning to 
the store of heated water with which the boiler-explosion 
began, at the point C, let it now be supposed that the boiler- 
walls did not give way, but that instead heat continued to 
be absorbed by the water. The temperature of the water 
being already in equilibrium with the pressure upon it, 
further advent of heat must develop isothermal vaporiza- 
tion. The entropy will increase along the isothermal CD 
(Fig. 9) and the volume will increase along the constant- 
pressure line CD (Fig. 10). At D vaporization is complete, 
and the steam is dry and saturated. 

In the actual engine, at this stage of the process, the 
steam leaves the boiler and passes to the engine-cylinder, 
where D represents in the actual engine the point of cut- 
off. Here the steam finds itself isolated from all heat-inter- 
change with the rest of the universe but free to impart its 
molecular energy to the piston in the shape of mechanical 
work, because the intensity of its kinetic energy exceeds 
the resisting pressure of the piston. 

Finding therein an opportunity to drop in temperature, 
it promptly does so, falling down the adiabatic DE 
(Fig. 9) in the process. At E further fall becomes 
impossible. In the case of the non-condensing engine, 
the limiting factor is atmospheric pressure ; in the case 
of the condensing engine it is sometimes the pressure 
and sometimes the temperature in the condenser ; that is, 
either a poor air-pump or a warm condenser will limit the 
vacuum. 

In order to leave the cylinder ready for a repetition of 
the process the steam in it must be condensed at this bot- 
tom temperature-level. The condensation therefore takes 
place under constant pressure, along the isothermals EB 
(Figs. 9 and 10), until it is complete at B. The pound of 
water is then ready for a repetition of the cycle BCDE. 



96 THE THERMODYNAMICS OF HEAT-ENGINES 

By the application of Equation 21 to any point of DE 
and BH in the same way as was done to the point H of the 
boiler-explosion the curves DE and BH of Fig. 10 can be 
developed. The cycle is then represented in the pressure- 
volume diagram by the same circuit BCDEB, in which CD 
and EB are isothermals and DE is an adiabatic. 

This cycle of processes is called the Cycle of the Perfect 
Steam-engine, or the Rankine Cycle. 

It will be noticed, first, that the adiabatic process DE, 
when represented in the pressure-volume diagram, results 
in a hyperbolic curve instead of a perpendicular drop. This 
shows that the adiabatic process is a heat-process, not a 
steam-process or a volume-process ; for the straight line of 
fall is the natural one. Indeed, the pressure-volume effects 
of adiabatic drop upon the various vapors and gases are 
the most diverse; whereas when viewed simply as heat- 
processes these diverse adiabatics always appear as vertical 
plumbline drops. 

In the second place, it should be noted that its effect 
upon the volume and pressure of steam-vapor is so complex 
as to defy exact mathematical expression. The curve DE, 
Fig. 10, is hyperbolic in character, as are all those near it, 
and asymptotic to the two axes ; but there is no known exact 
equation for it. The only exact, and the most convenient, 
method of deriving it is the graphical one through the 
medium of the corresponding curve in the entropy-tempera- 
ture diagram described above. 

Problem 13. What are conditions of operation and the 
efficiency of a perfect steam-engine operating between a 
boiler-pressure of 100 pounds per square inch by gage 
and exhausting into the atmosphere ? Represent its opera- 
tion by means of entropy-temperature and pressure-volume 
diagrams. 

The points B and C have already been located in con- 



THE STEAM-ENGINE CYCLE 97 

nection with Problem 12. The entropy of C appears from 
the tables to be 0.4860 and that of D, to be 1.5837. The 
entropy of E being the same as that of D and that of B 
being 0.31 16, the entropy BE appears to be 

1.5837 — 0.3116= 1. 2721. 

The specific entropy Be at this level being 1.4372, the 
ratio of the two entropies gives the steam-weight present 
as 0.8851 pound. The specific volume of steam at this 
pressure being 26.98 cubic feet, the volume of the wet 
steam after expansion to D must be 

0.8851 x 26.98 = 23.88 cubic feet. 

If the cycles of Problems 12 and 13 as portrayed in 
Fig. 9 be compared, it appears that the latter is made up 
of the triangle of the former, BCH, and the rectangle 
HCDE. The heat supplied in Problem 13 is greater than 
that of Problem 12 by the rectangle beneath CD; the heat 
rejected is greater by the rectangle beneath//^; the heat 
converted into work is greater by the rectangle HCDE. 
The heat supplied is 

<2i = l\ — 4 + A — 1004.4 B.t.u. 

The heat rejected is the entropy BE (= BH + CD) times 
the absolute temperature at E, or 

02 = W> - &b)T. = 855.0 B.t.u. 
Therefore 

Qw = Q x - (2 2 = IO °4-4 - 855-0 = 1494 B.t.u. 

The efficiency of conversion of heat into work, or the 
thermodynamic efficiency, is 

F= Qy-Q* = I 49-4 =I4-87 ^ 

Qx 1004.4 



98 THE THERMODYNAMICS OF HEAT-ENGINES 

If the area of the triangle which measures the heat con- 
verted into work in the boiler-explosion cycle be expressed 
analytically, there results 



k-k- TINo -N B ) = l x - / 2 - T 2 S log' 






If the area of the rectangle HCDE which is added to this 
in making up the Rankine cycle be expressed analytically, 
there results 

{T X -T^N -N^)={T X -T^. 

If these be added, there results 

Q* = h~l % -T % S log 6 p + L i(^f^} (22) 

This is the algebraic expression for the utmost work attain- 
able from dry steam between the limits T x and T 2 . It is 
inexact in that 5 is not a constant over any wide range of 
temperature. The calculation of the graphical areas by 
means of the steam-tables is therefore to be preferred, for 
accuracy as well as for convenience. There may arise 
occasions, however, when the above expression is useful. 
With a proper choice of value for S its inaccuracy is not 
great. 

The rectangle HCDE must represent a Carnot cycle, of 

T — T 
the maximum efficiency, given by the expression 1 — 2 > 

1 
The last term of Equation 22 is easily recognizable as 

arising from such a cycle, L 1 being the heat supplied along 
the isothermal CD and the parenthesis being the efficiency- 
coefficient by which it is reduced to work done. It will 
also be noted that the general contour and the efficiency 
of the Rankine cycle BCDEB closely approach those of 
the Carnot cycle HCDEH. In Problem 13, 14.88/0 of the 
heat was converted into work. Had the Carnot cycle 



THE STEAM-ENGINE CYCLE 99 

been followed between the same temperature-limits, the 
portion of heat converted into work would have been 

(798.3-672.1)^798.3=15.81^. 

The difference is not worth striving for. Here can be 
seen for the first time what will be frequently referred to 
again : That the inefficiency of the steam-engine is due 
neither to a poor cycle nor to an improper working-sub- 
stance, but to narrow temperature-limits. For its cycle is 
a close approach to the ideally perfect rectangle. 

Problem 14. If the engine of Problem 12 were fed with 
steam 20°[o wet, what would be the efficiency, etc. ? 

In this case the boiler-steam, instead of being completely 
evaporated to dryness at D, leaves the boiler with one-fifth 
of its weight still unevaporated water. Its condition is 
shown at the point F. The expansion now follows the 
adiabatic FG, and the complete cycle is BCFGB. At F its 
entropy is 0.4860 + (0.8 x 1.0977)= 1.3642; its volume is 
0.8 x 3.881 = 3.105 ; its total heat is 

308.05 +(0.8 x 876.35)= 1009.1. 
In this case 

Q x = 1 009. 1 — / 2 = 1009. 1 — 1 80. 1 =829.0 B.t.u. 

The entropy of vaporization at G is 

1.3642 —0.3116= 1.0526; 

the specific entropy of vaporization at this level being 
1.4372, the steam-weight present at G is given by the ratio 
between these two, or 

•—* — = 0.7324. 100—73.24=26.76. 
1.4372 * /3 

The initial 20% of water has become 26.76% during ex- 
pansion. 

In this problem Q 2 is the area of the rectangle under 



IOO THE THERMODYNAMICS OF HEAT-ENGINES 

BG, or 965.9 x 0.7324 = 707.42. Hence the work done is 
829.0—707.42=121.58 B.t.u., and the efficiency is 14.66%. 

In actual practice the deleterious effect of wet steam is 
much greater than that shown by a comparison of these 
two problems. The trouble is not that the water diminishes 
the theoretic possibilities for the development of work, but 
that its presence interferes with the realization of those 
possibilities, in a manner to be described in Part II. 

Problem 15. If the engine of Problem 13 were to ex- 
haust into a condenser maintaining a vacuum of 25" of 
mercury instead of into the atmosphere, what would be 
the efficiency, etc. ? 

Here the lower temperature-limit to which adiabatic drop 
may take place has been lowered by the decreased back- 
pressure to 130 F., or 590.8 absolute. This tempera- 
ture-level is represented by the line JK (Fig. 9). All of 
the quantities can be calculated as before. 

Problem 16. Find the results of exhausting the engine 
of Problem 14 into a vacuum of 25". 

Problem 1 7. If the engine of Problem 1 3 were to be fed 
with steam superheated to a temperature of 500 F., the 
pressures remaining the same, what would be its effi- 
ciency, etc. ? 

In Figs. 9 and 10 the point D represents steam just at 
the point of saturation, to be superheated only by addi- 
tional heat. The addition of this heat will increase both 
temperature and entropy according to the equation 

N-N D =S\og e ^r, 

J- D 

where the subscripts indicate the point D and the value of 
>S is the specific heat of steam, or about 0.55. This process 
is shown by the curve DX, X being the condition of the 
steam in the engine just before expansion begins. The 



THE STEAM-ENGINE CYCLE 10 1 

line DX in the P F-diagram would be a straight horizontal 
extension of CD, for the superheat causes the volume to 
increase under constant pressure. Both DX and the ex- 
pansion-curve XYZ are omitted from the P /^diagram, to 
avoid confusion, for the latter would be a hyperbolic curve 
lying close to and crossing .S5 at a very sharp angle. The 
efficiency and any volumes desired can be investigated 
and calculated with the help of the entropy-temperature 
diagram alone. 1 

When cut off from the boiler the steam-heat at X promptly 
drops in temperature down the adiabatic XY X.o Y, at which 
point no further drop is possible. The rest of the heat 
must be abstracted along YB. The increase in entropy 
due to DX is found from the above equation to be E Y= 
aio.20. Taking the value of 1.2 721 for BE from Prob- 
lem 13, BY=BE + EY= 1. 2721 +0. 1020 = 1. 3741. Since 
Be= 1.4372, the steam-weight present at Fmust be 0.9491 
pound. The Q x of this problem must be greater than that of 
Problem 13 by the heat added along DX=o.^ (500—337.4) 
= 89.4 B.t.u. Q 1 = 1093.8. The Q 2 must now be greater 
than that of Problem 13 by the rectangle under EY, or 
it may be determined directly as the area beneath BY; 
whence ^ = 923.5. Therefore Q w , the work done, is 
<2i — Q 2 = I0 93-8 — 923.5 = 170.3 B.t.u., and the efficiency 

of the cycle is = 15.6%. 

3 1093.8 ' 

The beneficial effect of superheat upon the efficiency in 

the actual engine is much more than what is shown by 

comparison of this result with Problem 13. The reasons 

for it are the exact reverse of those processes which make 

1 The close confusion of these several diverse curves in the P /^-diagram, 
although they are entirely distinct in the iVZ'-diagram, stands witness to the 
obscurity of the evidence as to what is thermally transpiring in the cylinder 
which is offered by the indicator-card. 



102 



THE THERMODYNAMICS OF HEAT-ENGINES 



wet steam so deleterious, and will be explained in the Sec- 
ond Part of this work. 

Problem 18. What would be the efficiency and condi- 
tions of operation of a direct-acting steam-pump working 
non-expansively between a boiler-pressure of ioo pounds 
by gage and the atmospheric pressure ? 



W/ 



c |WMNm^™ 




,D 


i^\ <^^ -— — - — 


U 


-Aw 


B f%~^R^-- 





— \e 


Kl 







Fig. ii 



Again, the point D (Figs. 1 1 and 12) represents the con- 
dition of the steam-heat as the supply from the boiler is 
cut off in the pump-cylinder. But now, instead of being 
allowed to expand adiabatically, the heat is abstracted at 
constant volume while the piston stands still at the end of 
the stroke. In the actual pump the exhaust-valve is opened 
at the end of the stroke to a separate condenser, but ther- 
modynamically the abstraction of heat by the condenser 



THE STEAM-ENGINE CYCLE 



103 



may better be imagined as taking place entirely within the 
cylinder. 

The heat is now not free to fall naturally down the adia- 
batic DU, but is diverted artificially by the limitations of 
the case into following the path DM, called the constant- 
volume curve, the heat meanwhile dripping from the vari- 
ous points on this curve down to the tail-race level Be by 
means of " free fall." The curve DM has no known exact 




Fig. 12 



equation in the AT-diagram ; in the P ^diagram its equa- 
tion is V— a constant. But the AT-curve may be found 
by determining individual points by the help of Equation 21. 
In this case it is the volumes which are known and the other 
variables which are to be found. Thus, at M the volume 
is the same as at D, or 3.881 cubic feet. At this level the 
specific volume is 26.98 cubic feet. Therefore there must 

3.881 



be present a steam-weight of 



26.98 



0.1438 pound. Its 



104 THE THERMODYNAMICS OF HEAT-ENGINES 

entropy of vaporization is o. 1438 x 1 .4372 = 0.2067. Its total 
entropy is 0.5182, which locates the point M in the NT- 
diagram. Intermediate points on the curve DM, Fig. 11, 
may be found similarly. 

In this problem the Q x is the same as in Problem 13, or 
10O4.4 B.t.u. Q 2 cannot be found directly from the NT- 
diagram, but can be deduced from the value of Q w given 
by the P J^-diagram. The area of the rectangle BCDM, 
Fig. 12, is 100x144x3.881 = 55886 foot-pounds. 1 Therefore 

the area of BCDM, Fig. 11, is 1!^= 71 .8 3 B.t.u. This 

71.83 77 * 

shows an efficiency of = 7.1%. 

J 1004.4 

Problem 19. What would be the efficiency, etc., were 
the pump of Problem 18 to exhaust into a vacuum of 25" ? 

Here the cycle describes the path JCDKJ. 

Problem 20. What would be the efficiency, etc., of the 
engine of Problem 13 if, instead of carrying its expansion 
completely to atmospheric pressure, it should have reached 
the end of its stroke by the time the steam had expanded 
to a gage-pressure of 20 pounds and the exhaust-valve 
had then been opened ? 

The cycle now describes the -path BCDURB (Figs. 11 
and 12), the w«-level being at the temperature corresponding 
to saturation at 20 pounds by gage, or 258. 2° F., or 719 
absolute. The cycle may be divided into two distinct por- 
tions : (1) wCDU and (2) BwUR. The first portion can 

1 In calculating quantities of work from volumes and pressures of fluids, 
whether the latter be steam, water, air, or other gases or liquids, it is engineer- 
ing custom for volumes to be always measured in cubic feet and pressures in 
pounds per square inch. At the same time the resultant work is customarily 
desired stated in foot-pounds. A slight consideration of the question will 
make it plain that the only pressure which can be multiplied by volumes in 
cubic feet, with the resultant product in foot-pounds, is one measured in 
pounds per square foot. The factor of 144 must therefore be entered into all 
such equations to translate pounds per square inch into pounds per square foot. 



THE STEAM-ENGINE CYCLE 105 

be treated similarly to Problem 13 ; in fact, its Q x is the 
same. The second portion can be treated as Problem 18. 
For the entire cycle the Q w is obviously equal to the sum 
of the Q w 's for the two portions. The Q 2 , if desired, may 
be found from the Q 1 and the Q w . 

Problem 21. What would be the efficiency of the engine 
of Problem 20 if it were to exhaust into a vacuum of 25" 
instead of into the atmosphere ? 

The cycle now describes the path JCD URIJ. 

Wire-drawing. The Curve of Constant Heat 

Let A and B, Fig. 13, represent two steam-chambers or 
pipes in which the pressure is maintained at P and/ pounds 
per square inch respectively, connected by a passageway of 
relatively much smaller diameter, through which the steam 
finds its way from A to £ with comparative difficulty. This 
passageway may be a partially opened throttle-valve or 
blow-off valve, or it may be a long line of pipe too small 
for its task, or it may be the aperture in the diafram of a 
Peabody throttling-calorimeter. In any such case it is 
only necessary, in order that the following discussion shall 
apply, that (1) the pressure in B be less than in A, and 
(2) that the velocities in A and B be small in comparison 
with that in the passageway. 

Under these conditions let Q, Fig. 14, represent the 
heat-condition of a steam-and-water mixture present in 
the chamber A. The diagram shows it to be about 3% 
wet ; but this is only to illustrate a single application, that 
of the throttling-calorimeter used with commercial steam. 
It might be quite dry, it might be superheated, or it might 
be all water; the argument would apply equally well. 

As the steam leaves the chamber A, it finds itself under 
a decreased pressure. As pressure is all which keeps up 



io6 



THE THERMODYNAMICS OF HEAT-ENGINES 



the temperature, its diminution allows the temperature to 
drop ; it therefore starts down the adiabatic Qd. This 
process, however, even when started upon by only an 
infinitesimal degree, must develop mechanical energy in 
corresponding amount. There being no other form for 
this energy to take under the conditions, it takes the kinetic 
form ; or, in other words, the elastic condition of the steam, 
due to its heat, sets up a flow toward the lower pressure. 

The existence of flow immediately engenders friction, 
and friction is nothing more than a name for the trans- 
formation of kinetic energy into heat. By this double 













B 




A 


£ 


















p 










Fig. 


13 


£' 


P 


>■ 



















energy-transformation, therefore, the heat-energy disap- 
pearing in the first infinitesimal drop in temperature, 
which would be measured by the area of the rectangle 
of infinitesimal height running from CC X out to Q, must 
immediately reappear as heat which is absorbed by the 
steam which developed it. For no energy can be lost, 
either as heat or as work; all must be returned to the 
steam by friction, eddy, or impact. 

The return of this heat, at the instantaneous pressure 
then prevailing, appears in the form of isothermal vapori- 
zation along the line C X Q X to an infinitesimal amount and 
must be measured by the area under the process-curve. 
This curve is a horizontal line dN in length. The area 



THE STEAM-ENGINE CYCLE 



IO7 



beneath it is the rectangle of infinitesimal width Q x d x d, 
the area of which is T x dN. 

By a series of such infinitesimal steps is developed the 
curve Q\Q 2 Q^q- The law which defines it is that the heat- 
energy apparent at any point must be equal to that at any 
other point, because equal to that originally present at the 




Fig. 14 

point Q. This may be stated geometrically by saying 
that the area under the constant-pressure line to any point 
of the curve, such as the area beneath AC X Q V AC 2 Q 2 , 
A C z eQ z , A C±kQ A , etc., must be equal to that at the original 
point; for the integration of the horizontal and vertical 
infinitesimal rectangles to equal limits must result in equal 
areas. C 2 C 1 Q 1 Q 2 is therefore equal to Q 1 d 1 d 2 Q 2 . Sub- 
tracting the area Q x mQ 2 from each side and adding the 



108 THE THERMODYNAMICS OF HEAT-ENGINES 

area, OAC< l md l gives OAC x Q x d x = OAC 2 Q 2 d 2 , which is 
merely the graphical statement of the second sentence 
of this paragraph. 

The heat-quantity of the point Q v which is also the heat- 
quantity of every other point in the curve Q 1 Q 2 (?, is called 
the characteristic of the curve. There are of course as 
many constant-heat curves on the diagram as there are 
different quantities of heat imaginable. 

It is important to deduce the equation of the constant- 
heat curve. 

The simplest case is where the curve originates in the 
liquid-line A W, as at C v and develops the course C x F^F b , 
etc. Let F 3 be any point and let the subscript 3 be omitted 
from the algebraic notation. From the nature of the curve, 
the area c z C z C^f^_ must equal the area c z C z F z f z , or 

S ( Tl -T)-TS log e |i= T(N- NJ, 
whence N- iV, = s(^ - 1 - log, ^ 

Differentiating, dN= Sj$(T - T^'dT. (38) 

The area f\C x F z f z is made up of a summation of ele- 
mentary vertical strips each having the width <aWand the 
height 71 Therefore this area may be expressed mathe- 

TdN. Substituting the value just 

found for dN, integrating and reducing, 

area f x C x FJ z = S(T 1 - T 8 ) - 7\5 log, -f 

= area c % C z C x f x - area c z gC x f x = - area C z gC v (39) 

(The algebraic sign, in thermodynamic work, indicates 
merely the direction in which the heat is travelling in rela- 
tion to the form of energy portrayed by the diagram or 



THE STEAM-ENGINE CYCLE IO9 

equation in question. In final results, therefore, it may 
usually be neglected.) 

Stated in general terms, this result just demonstrated 
becomes : — 

When the energy involved in any liquid isomorphic drops 
in temperature by wire-drawing, the area beneath the con- 
stant-heat curve and between ordinates is equal to that be- 
tween the original isomorphic, the abscissa through its upper 
end and the ordinate through its lower end. 

The Peabody Calorimeter. — In this instrument, invented 
by Prof. Peabody, the chamber A of Fig. 13 consists of a 
steam-pipe, etc., from which a sample of steam is to be drawn 
for the purpose of determining its moisture and heat-value. 
The small passageway is usually represented by a small 
hole in a diafram, but it may be a slightly opened valve or 
a length of very small, well-lagged pipe without affecting 
the thermodynamics of the question. The chamber B is 
made large enough to let the steam come practically to rest 
and is equipped with a thermometer-well and with every 
possible precaution against the radiation of heat. It is best 
made freely open to the atmosphere, the pressure within it 
being determined by barometer ; but it may be closed from 
the air by a valve, if preferred, in which case a pressure- 
gage must be applied. The only data necessary for an ob- 
servation are (1) the pressure in A, (2) the pressure in B, 
and (3) the temperature in B immediately opposite the 
steam-entrance. The first gives the level of the C-^Qyline 
of Fig. 14, the second the level of the Q^-line, and the 
third the elevation of the point Q s , on the line ee'. Alge- 
braically, 

k +J>L 1 = & S + o.55(4-^ 3 ), 

wherein the quantities in the first member apply to the 
observed pressure in A, and those of the second to the 



110 THE THERMODYNAMICS OF HEAT-ENGINES 

barometric or other pressure in B. The unknown propor- 
tion of steam in the steam-and-water mixture at Q x is rep- 
resented by y. The observed actual temperature in the 
chamber B is represented by t a . The temperature of satu- 
ration corresponding to the observed pressure in B is / 3 . 
In this equation/ is the only unknown quantity. 

Problem 22. Under a barometric pressure of 29.3", 
steam is drawn from a pipe under a pressure of 93 pounds 
by gage and shows in the calorimeter a temperature of 
228 F. What was its original percentage of moisture ? 

Problem 23. In the common marine practice of drawing 
steam from the boilers at a pressure of 300 pounds by gage 
and throttling at the engine to 250 pounds, how much of 
an original 2 % of moisture in the steam as it left the boiler 
will reach the engine, radiation being neglected ? 

The Fireless Locomotive. — This type of street-railway 
motor was widely experimented upon about 1870- 1880 in 
France, and somewhat also in this country. It consisted 
of wheels driven by an ordinary pair of locomotive engines 
and carrying a plain tank fit for an internal pressure of 
about 300 pounds to the square inch. At the start this 
tank was nearly filled with water. Into the water steam 
was blown from a high-pressure boiler until the water was 
brought to the saturation-temperature of the boiler and 
would condense no more steam. From the top of the tank, 
or from a dome, steam was drawn through a reducing- 
valve to supply the engines. As the draft of steam re- 
duced the pressure in the tank, the heat of the water 
vaporized a portion of itself into steam, and thus the 
engines were driven, the temperature in the tank steadily 
dropping until its pressure was no more than sufficient to 
drive the engines, when it must be recharged. The fireless 
locomotive never reached practical success, by reason of 
its weight and of the early appearance of its much more 



THE STEAM-ENGINE CYCLE III 

agile rival, the electric trolley-car. But it presents a none 
the less interesting problem in thermodynamics, which 
represents a wide class of actual situations arising wherever 
hot water is stored and drawn upon through a reducing- 
valve for a supply of steam. 

Problem 24. If a fireless locomotive have a cylindrical 
tank 6 feet in diameter by 1 5 feet long, with spherical ends 
of 6-foot radius, filled with water heated to a pressure of 
300 pounds by gage ; if the pressure be reduced to 80 
pounds by gage at the engine ; if it requires 40 horse- 
power to move the train at 15 miles per hour and the gen- 
eral efficiency of cylinders and connecting gear be 50^0; 
how far will the locomotive travel before recharging be- 
comes necessary ? (The exhaust is of course atmospheric.) 

Referring to Fig. 14, let C x be the 300-pound tempera- 
ture-level, let C s be the 80-pound temperature-level, and let 
C 4 be the atmospheric-pressure temperature-level. Then 
the heat in each pound of water when the locomotive is 
freshly charged will be measured by the area OAC x f v 
When the locomotive has travelled until its tank-pressure 
has fallen to 80 pounds by gage, each pound of this water 
contains the heat OAC s c s . To be sure, much of the origi- 
nal weight of water has left the boiler during the trip ; but 
as is always the rule, locality does not count. Wherever 
the water may be, in order to get out of the boiler it must 
have had at least the latter amount of heat per pound. 

The difference between these two heat-quantities, or the 
area c.§C z C x f v is the energy which has left the tank through 
the reducing-valve during the trip. 

Of this energy the portion C 3 C x w converts itself, by 
wire-drawing through the reducing-valve and the conse- 
quent isothermal vaporization, into the area/jwi^/g. The 
total steam-heat which reaches the cylinders is, therefore, 
the area c s C 3 F 3 f s . Of this, only that above the Q-level of 



112 THE THERMODYNAMICS OF HEAT-ENGINES 

atmospheric exhaust is available for doing work. In other 
words, the heat which represents the work available for 
drawing the locomotive is the area nC s F 3 r. 

From the steam-table, the temperature at C s is 784.3, 
and the heat of the liquid is 293.58. At C x the tempera- 
ture is 882.45, and the heat of the liquid is 396.44. The 
area c 3 C z C 1 f 1 = 396.44 — 293.58 = 102.86 B.t.u. This is 

also equal to the area c a C s F 3 f 3 . Of it only proportion — 3- 
is available for work, or 3 3 

nC s F B r= 784 - 3 ~ 672A - 102.86=14714 B.t.u=i 1,447 ft.-lb. 

At the start the tank contains 23,680 pounds of hot 
water. The efficiency of transmission is 50 f>. To move 
the train 1 5 miles requires 40 horse-power for one hour, or 
79,200,000 foot-pounds. Therefore the distance which the 
locomotive will travel, with the train attached, before its 
tank-pressure falls to 80 pounds by gage, is 

23,680 X H,447 X °-5 H H ^ -i 

-^ imu 2_ . 1$ — 25.6 miles. 

79,200,000 

Problem 25. When the blow-off cock of a boiler oper- 
ated at 80 pounds pressure by gage is opened, what pro- 
portion by weight of the issuing current is steam ? 

The foregoing illustrations cover, so far as the author is 
aware, the ultimate limiting theory of every problem which 
may possibly arise in any form or variety of steam-engine 
or of steam-apparatus. 



CHAPTER V 

THE LAWS OF THE PERMANENT GASES 

All forms of matter occur in solid, liquid, or gaseous form, 
according to the pressure and temperature to which they 
are subjected. With most of them the change from liquid to 
gaseous is more or less gradual. As heat is added there is 
first evolved a form of matter called a vapor, of which ordi- 
nary steam is a good example. Upon further heating this 
is superheated and gradually becomes a perfect gas. The 
vapors obey many of the laws of the gases and are distinctly 
similar to them in their characteristics ; but in some points 
they are quite different, notably in their action under heat. 
The gases, on the other hand, are distinguished from the 
vapors by no hard and fast line. They are merely highly 
superheated vapors. 

It is to be noted most emphatically, however, that the 
laws controlling the mixtures of liquids and their saturated 
vapors bear a strong contrast to those of the highly super- 
heated vapors called the "permanent gases." This con- 
trast or distinction, although developed by a differentiation 
which is gradual, is broad and wide. It must never be lost 
sight of in thermodynamic practice. In the preceding 
chapter attention was centred upon the laws governing 
the effects of heat upon liquids, saturated vapors, and 
mixtures of liquid and vapor ; in this chapter are discussed 
the quite different laws governing the action of heat upon 
the permanent gases. The laws of the heat itself, under- 
lying both, are invariable. 

i 113 



114 THE THERMODYNAMICS OF HEAT-ENGINES 

Boyle's Law. — When the temperature of a gas remains 
constant, the pressure must vary inversely with the volume. 
Expressed mathematically, this reads : — 

^ = ^- (23) 

P V y 5) 

Charles's Law. — When the volume remains constant, 
the pressure must vary directly with the absolute temperature. 
Expressed mathematically, 

P T 

-p=-r> ( 2 4) 

where the zero subscript refers to the original temperature. 
Suppose the original condition of a gas to be the press- 
ure P' , the temperature T' , and the volume V . Suppose 
that, while the temperature is kept constant at T', the 

volume is changed to V" . The new pressure, p, must 

pi y 
equal • Suppose, again, that, the volume remaining 

constant at V" , the temperature be altered to T". The 
latest pressure, P" , must equal p — -, whence 

D „ P'V'T" P'V P"V" . . ,,. 

P = V n T , > or ~jT = ^f^ = a COnstant (25 ) 



From this, if the pressure be constant, 
K To 



(26) 



The nature of the constant of Equation 25 can be seen 
from a consideration of the origin of the idea of the abso- 
lute zero of temperature ; viz., that, since gas at 32 F. in- 
creases ^^g of its volume for each degree of alteration in 
temperature, a reduction in temperature of 493 must result 
in a reduction of the volume to zero, beyond which there 
could be no further abstraction of heat. Any volume of 



THE LAWS OF THE PERMANENT GASES 115 

gas may therefore be imagined as having been derived from 
heating a zero volume from absolute zero of temperature to 
the given one. During this process the external work per- 
formed must be equal to P V. The external work done per 

P V 
degree change in temperature must be equal to — =-• But 

this is the constant of Equation 25. Therefore : — 

The relation of pressure, volume, and temperature of a 
given mass of any gas at any instant is given by the equation 

%- C, (V) 

in which the value of the constant is the number of foot-pounds 
of external work done during an alteration of temperature 
of one degree under constant pressure. 

The volume of one pound of air at 32 F. under a press- 
ure of 14.69 pounds per square inch is 12.39 cubic feet. 

12.39x14-69x144 ^ 53 . l8 foot-pounds. 
492.8 

Therefore it always holds true for one pound of air that its 
pressure in pounds per square inch times its volume in 
cubic feet is 53.18 times its absolute temperature in 
Fahrenheit degrees, or for air, 

PV= 53-18 7: (28) 

The specific volume of air is usually stated in the hand- 
books as a certain volume at a certain temperature under a 
certain pressure. This set of three figures it is impossible 
to retain in the memory. The single figure, 53.18, however, 
can easily be so retained. From it the volume of a known 
weight of air under any given conditions can be calculated, 
and more readily even than from the data given in the 
hand-books. 



Il6 THE THERMODYNAMICS OF HEAT-ENGINES 

A slight consideration of the method of derivation of 
the significance and the value of this constant will show 
that for other gases than air it must be inversely propor- 
tional to the density of the gas in question. From a table 
of specific gravities and the figure 53.18, therefore, can be 
readily calculated the volume of a known weight of any gas 
under any given conditions. 

Specific Heats of Gases. — When gas is heated, one pound 
of it does not always absorb the same amount of heat per 
degree rise in temperature, independently of conditions, as 
is approximately the case with water, etc. If the gas be 
heated while the volume is kept constant, it is obvious that 
the only energy absorbed is that required to impart to the 
vibrating molecules the increased velocity which expresses 
itself to our senses as an increase in temperature. On the 
other hand, if the same gas be heated to a like degree 
under constant pressure, not only must this same energy 
be supplied in order to increase the velocity of its molecu- 
lar vibration, but also its volume will increase and external 
work must be done in overcoming the external pressure 
through this increase in volume. The heat absorbed in 
increasing the internal, vibratory energy of the molecules 
is the same in either case, otherwise the temperature- 
increase would not be the same. The external work of 
the latter case is quite additional to this and is peculiar to 
the condition of pressure kept constant. 

If S v and S p be the specific heats under constant volume 
and constant pressure respectively, then, since the constant 
C of Equation 27 is the external work done in one degree 
rise in temperature under constant pressure, 

S p= S v + Z^> ( 2 9) 

77% 

(since the specific heats are measured in B.t.u., and the C in foot- 
pounds). 



THE LAWS OF THE PERMANENT GASES WJ 

If K be the ratio between the two specific heats, then 

K=^ (30) 

^ St= 778(^-1)' (3I) 

A table of the values of S\„ S p , and K for the gases most 
commonly met with in engineering problems will be found 
in the Appendix. It will be noticed that they all lie 
fairly closely together, with the exception of hydrogen ; 
and this gas, in spite of having a specific heat some four- 
teen times as great as the others, has a value of K very 
similar to the rest. 

Expansion-curves, /'/-diagram. — When a gas expands 
without special means being taken to maintain either 
pressure or temperature constant, its pressure and vol- 
ume change according to some law which usually can be 
expressed by the equation P V x = a constant, or 

the latter usually being the more convenient form for use 
in calculation. When air or similar gases expand or are 
compressed in the cylinder of a motor or compressor, x 
may assume any value between unity and about 1.40. 

When a gas expands in this way, the external work done 
is that involved in pushing back the resisting pressure P 
through the distance represented by the increase in volume. 
But P not being constant, the amount of external work 
done is not directly observable. 

Let the curve AB of Fig. 15 represent the pressure- 
volume changes of a gas expanding or being compressed 
according to Equation 32. The external work done dur- 
ing an infinitesimal amount of expansion, at any point, as 



u8 



THE THERMODYNAMICS OF HEAT-ENGINES 



c, must be the area of the elementary rectangle cv, or PdV. 
The total external work done in expanding between A and 
B must be the integration of this expression, or the area 
V 2 BA V v or 



PdV=P 2 V 2 x \ — = P 2 V 2 *- i-i il- 

r 2 »/k V x i —x 



^ P2^2-PiV^C(T 2 -T,) 



x — I 



X — I 



(33) 



This is the external work actually performed in a mere 
change of state from the condition P 1 V x to the condition 




Fig. 15 



In the actual expansion or compression of air or gas in 
the cylinder of a motor or compressor, however, each 
revolution of the machine involves more external work 
than merely that of the alteration in pressure and volume ; 



THE LAWS OF THE PERMANENT GASES 119 

there is external work involved in first drawing the gas 
into the cylinder before the transforming process begins, 
and again in afterwards discharging it from the cylinder. 
These quantities are in addition to that discussed above. 
Such a triple process is shown in Fig. 15 by the area 
P 2 BAP V In this the rectangle P 2 B V 2 is the external 
work of taking the gas in (if the machine be a motor), and 
P X A V 1 is that of discharging it. Of these two quantities, 
the former always has the same sign as the quantity of 
Equation 33 ; the latter always has the opposite sign. 
Adding these two algebraically to Equation 33, or, what 
is the same thing, integrating the horizontal elementary 
rectangle pc, whose area is Vdp, to the limits P 1 and P 2 , 
there results, as the work done between abscissae, 

it/ p 2^2- p i v i C ( T z- T i) / x 

W « = x-i = Y-i • <34) 

x x 

Equation 33 gives the true measure of transformation of 
heat into work during the expansion, or of work into heat 
during the compression, of a permanent gas ; it includes 
no other process than the mere alteration of condition of 
the gas. Equation 34 gives the measure of the work 
actually given out by a motor, or absorbed by a com- 
pressor, in handling a perfect gas ; it includes, in addition 
to the former process, the displacement of the gas from a 
locality of initial to one of final pressure. 

The alteration in temperature resulting from compres- 
sion from A to B, or from expansion from B to A, can be 
known by combining Equations 27 and 32. 

Ii = p^Zi = YlXi = (F2Y- 1 = (PiY ( * 
r 2 p 2 v 2 v 2 v{ \vj v/y ' KSiJ 

It will be of mnemonic aid to note that the denominator 
of the expressions for the external work are x — 1 when 



120 THE THERMODYNAMICS OF HEAT-ENGINES 

referred to the voltime-axis and when referred to the 

x 

pressure-aids, ; and similarly, that the exponent of the ex- 
pression for the alteration in temperature is x — I when 

referred to volumes and is when referred to pressures. 

x 

In making arithmetical calculations of the quantities 
handled by these equations it will be found a great time- 
saver to keep them always in the form given in Equations 
32 and 35 ; that is, in the form of ratios of initial and final 
conditions. In doing this, if the subscripts be chosen so 
as to make the value of the fraction greater than unity, the 
logarithms will always be positive and so lend themselves 
to the raising to odd powers with much greater con- 
venience than if otherwise. 

Equations 32 and 35 might be stated in the form; 

or Equation 25 might be stated in the form, 



p* v x p 



From either may be deduced the very convenient rule that 
the logarithm of the temperature-ratio involved in a given 
amount of compression or expansion is always equal to the 
difference between the logarithm of the pressure-ratio and 
that of the volume-ratio, when such ratios are all kept 
greater than unity. 

Problem 26. An air-compressor handles 100 cubic feet 
per minute of atmospheric air at a temperature of 69 F., 
compressing it to 60 pounds gage-pressure according to the 
equation : P V 1%s = a constant. What is the final volume and 
temperature of the air and how much power is absorbed? 



THE LAWS OF THE PERMANENT GASES 121 

P x = 14.5 log/ ? 1 = 1.161368 

P 2 = 74.5 log P 2 = 1.872156 p ^ 

Subtracting, 0.710788 = log — 2 

V l 

Dividing by 1.3, log — f = 0.546760 

^2 



T 



Subtracting, o. 1 64028 = log — 3 

A 
7^ = 66 + 491 = 530 log 7\ = 2.724276 

Adding, 2.888304 = log T 2 

T 2 = 773.2 absolute. t 2 =312.2 F. 

V 1 = 100 log Fj = 2.000000 

tog T7 = o-54676o 

2 1.453240= log F 2 

F 2 = 28.39 cubic feet. 

P 2 V 2 - P x l\ 

x — 1 



The Work Absorbed = 



x 
= [(7+5 x 28.39) -(14.5 x 100)] 144 

0.3- 1.3 
= 446,000 foot-pounds (per minute) 
= 13.5 horse-power. 

This sample calculation includes every digit essential to 
the solution of the problem, except that the arithmetical 
computation of the last equation (for the larger numbers in 
which logarithms are already noted) is omitted. 

Adiabatic Expansion of Gases in the /'/-diagram. — In 
adiabatic expansion the external work done must be at the 
expense of internal energy ; there is no other source whence 
it may come. In the perfect gases the internal energy is 
all of the temperature or molecular-velocity form, and its 



122 THE THERMODYNAMICS OF HEAT-ENGINES 

specific amount per degree alteration in temperature is the 
specific heat at constant volume, S v . Whence, if the gas 
be expanded from a temperature T x to T 2 , the internal 
energy lost or gained must be 

But, from Equation 33, the external work done during ex- 
pansion between those same temperatures is, if expressed 
in B.t.u., „,„, _ 

c(T A - r t ) 
7780-1)' 

If the expansion be adiabatic, these two quantities must be 
equal, whence x = K. Therefore, the P V-equation for the 
adiabatic expansion of a perfect gas is 

' 9AW- ^ 

Isothermal Expansion of Gases in the /'/-diagram. — If 

the expansion be isothermal instead of adiabatic, T becomes 
a constant and Boyle's law prevails. The equation of the 
curve AB, Fig. 15, becomes PV=sl constant, and an 
analysis similar to that by which Equation 33 was derived 
gives as the external work done during the expansion or 
compression 

W=P 1 V 1 log, ^ = P,V 2 log, ii (37) 

The work done between horizontals, representing the 
case of the actual motor-cylinder, is in this case the same 
as that between verticals. 

Regnault's Law. — It was announced by Regnault that 
the specific heat of gases, taken under constant pressure, 
is the same for all temperatures. This is so nearly true 
that it will be accepted as such for all engineering pur- 



THE LAWS OF THE PERMANENT GASES 1 23 

poses. But the fact that it is not quite true shows it to be 
a statement of coincidence rather than a law. 

Joule's Law. — It was announced by Joule that when a 
gas is allowed to expand without doing work its tempera- 
ture remains constant. This experiment was tried by plac- 
ing two vessels containing air at different pressures in a 
tub of water until all temperatures were alike. Communi- 
cation was then opened between the two. The gas in the 
vessel under heavier pressure expanded into the other; 
but no work was done, the system being considered a unit. 
The water revealed no alteration of temperature in the air. 

The explanation of the result is that all of the internal 
energy of the gas consists of kinetic molecular energy, or 
sensible heat, the disgregation work being zero because of 
the wide separation of the molecules of a gas one from 
another and their consequent small attraction for one 
another. But just because their mutual attraction, accord- 
ing to the law of gravitation, can never be quite reduced 
to zero by separation, though easily made imperceptible, 
it is obvious that the disgregation work even of a gas 
must be some positive quantity ; and if so it must be sup- 
plied during expansion, no matter whether external work 
be done or not, at the expense of the temperature-heat. 
Some change in temperature must result. This was proven 
by Lord Kelvin in some very delicate experiments upon 
the flow of gas through a porous plug. The fact has 
become of importance in the industrial arts within a very 
few years by the invention of the modern method for the 
production of liquid air, which depends entirely upon the 
absorption of temperature-heat in disgregation work when 
air expands without doing work. 

That this matter may be given further consideration, let 
attention be turned to Fig. 14, which is reproduced here from 
page 107. As the curve Q 2 q departs farther and farther from 



124 



THE THERMODYNAMICS OF HEAT-ENGINES 



the saturation-curve 55 it will plainly become more and 
more nearly horizontal and straight. It must be asymp- 
totic to ON. At a great distance away from 56" it must 
represent the condition of a very highly superheated vapor 
at a very low pressure. With steam, which Fig. 14 has 
been drawn to represent, these conditions would be so 




Fig. 14 

extreme as never to be found in actual occurrence upon 
the earth's surface. But if Fig. 14 be used to illustrate 
the thermal condition of one of the permanent gases, or of 
air, with which vapor-pressures are very much higher for 
a given temperature, and vapor-temperatures ^very much 
lower for a given pressure, than with steam, such a distant 
point to the right of 55 might well represent the thermal 
energy of air under quite familiar conditions. 



THE LAWS OF THE PERMANENT GASES 12$ 

If two adjacent points, both quite distant from vSvS upon 
the now nearly horizontal constant-heat curve Q 2 g, be con- 
sidered, it is plain that transition from one to the other 
must involve alteration of pressure, since Q 2 q crosses all 
of the constant-pressure isomorphs. But drop in pressure 
without alteration in total fund of heat was the process 
with which Joule experimented in establishing his so-called 
law. It is now evident, from the graphical argument 
involved in Fig. 14, that such a process could not possibly 
take place without drop in temperature, because following 
the curve Q 2 g, which is asymptotic to ON. But Fig. 14 at 
the same time reveals the reason why Joule noticed no 
drop in temperature ; the curve Q 2 q had become so nearly 
horizontal that the maximum variation in pressure possible 
developed a drop which was at that time imperceptible. 

It is now plain, in the light of accumulated knowledge, 
that Joule's free expansion of air was necessarily an illus- 
tration of the second law of thermodynamics. It could 
not have taken place, there would have been no motive- 
power to initiate it, had there been no concomitant drop in 
temperature. Drop of energy in intensity is the sole 
motive-power for all natural phenomena. 

Liquid Air. — In Fig. 16 let the curves W and S rep- 
resent the entropy-temperature relations of liquid air and 
saturated vapor of air respectively. Since the critical tem- 
peratures of oxygen and nitrogen nve — 18 1° and — 23 1° F. 
respectively, or 280 and 230 degrees absolute Fahrenheit 
respectively, the entire diagram must be imagined as lying 
below these very low temperatures. The boiling-points 
under atmospheric pressure of oxygen and nitrogen are 
165 and 1 44 F. absolute respectively. Let CCC be the 
constant-pressure curve at atmospheric pressure. Let AA 
and BB be two isothermals of vaporization at pressures 
considerably above atmospheric ; in practice they are com- 



126 



THE THERMODYNAMICS OF HEAT-ENGINES 



monly 30 and 10 atmospheres respectively. Let A A' and 
BB' be isomorphs of superheated air-vapor at these same 
pressures respectively. The ordinary thermal condition of 
atmospheric air would then be represented by some such 
point as C . 

In Fig. 17 let C be the diagram of an air-compressor 
taking in atmospheric air and compressing it, as nearly 



S.00°Ab.solute.= 39 Fahr. 



w s 



A/ 200 "Absolute £■ 
R^L 180° 




Fig. 16 

isothermally as possible, to the pressure of BBB' , Fig. 16, 
and discharging it into the reservoir B, Fig. 17. Let A 
be another compressor taking its supply of air from B 
and discharging it into the conduit aaa at the pressure of 
AAA', Fig. 16. This high-pressure conduit aaa passes 
into the interior of the larger pipe bbb. The two together 



THE LAWS OF THE PERMANENT GASES 



127 



are formed into a coil presenting a large amount of sur- 
face and leading to an end at V. The coil is encased on 
all sides with thermal insulation III, which is constructed 
with great care. At the terminus of the pipe aaa is a 
valve V opening into the larger medium-pressure pipe bbb 
and fitted to be adjusted by hand from without. From the 
terminus of bbb is a drain-line passing downwardly through 
the insulation and ending in a drain-cock D. 




Fig. 17 



The thermal activity of the apparatus illustrated in 
Fig. 17 may now be traced by means of Fig. 16. 

The compressor C supplies air at atmospheric tempera- 
ture to the reservoir B at the same temperature. The 
compressor A supplies air to the pipe aaa at the higher 
pressure AAA 1 , but still at atmospheric temperature ; for 
temperatures may be supposedly kept down to atmospheric 
by cold-water jackets. This double process may be shown 



128 THE THERMODYNAMICS OF HEAT-ENGINES 

in Fig. 16 by the line C'B'A', of isothermal compression 
(although properly at a higher level than drawn). The 
highly compressed air then traverses the pipe aaa and 
arrives at the valve V in the condition A', Fig. 16. It 
finds the valve V slightly open and is wire-drawn through 
it into bbb, its pressure falling to BBB' and its temperature 
to b 2 , Fig. 1 6, by passage down the constant-heat curve 
A'b % . 

In traversing the pipe bbb, Fig. 17, this current of air 
brings the ^4-pressure air coming in along aaa down to its 
own temperature, so that the next quantity of the latter to 
arrive at V finds itself there in the thermal condition a 2 , 
Fig. 16. Its wire-drawing through the valve f must then be 
represented by the constant-heat curve a 2 b s . The ^-press- 
ure air then traversing the pipe bbb at the temperature b s 
necessarily brings the next incoming air to the tempera- 
ture a s , and its wire-drawing in turn develops i?-pressure 
air of the condition b±. 

Lord Kelvin's formula for the drop in temperature D 
involved in each of these passages across from the AA'- 
isomorph to the i?i?'-isomorph is 

D = o. 497 (P A -P B )(^J, (41) 

wherein the pressures are measured in atmospheres and 
the temperatures in degrees Fahrenheit. T is the initial 
absolute temperature at the departure from the upper 
pressure. 

So the process goes on, all temperatures within III fall- 
ing steadily meanwhile, until finally the steady fall of the 
point successively represented by b 2 , b s , b it etc., results in 
its coincidence with B. As the processes which resulted 
in temperature-fall due to abstraction of heat still continue, 
their further action must result in the condensation of some 
of the air at the i?2?-pressure. This condensation drains 



THE LAWS OF THE PERMANENT GASES' 1 29 

to the valve D, and may be there drawn off into an external 
vessel. 

It will be obvious, from a slight consideration of the 
geometrical laws controlling the curves SB and ab, that at 
these very low temperatures they must be much nearer to 
parallelism than is the case at ordinary steam-temperatures. 
How nearly they approach parallelism, whether a reaches 
A before b reaches B, or the opposite, cannot now be said. 
If a does reach A before b reaches B, it must indicate that 
liquid is first formed in the pipe acta, just above the valve 
V, before it collects at D. If so, this liquid is vaporized 
by wire-drawing through V just as the moisture in a 
throttling-calorimeter is evaporated by a similar process. 
But whether this be the case or not does not affect the 
thermodynamic explanation of the final result. 

Carnot's Cycle with Air as a Working-Substance. — 
Carnot's cycle, consisting as it does of two isothermals and 
two adiabatics, must always appear as a rectangle upon 
the TVT-diagram. But the form of its P /^-diagram de- 
pends entirely upon the working-substance which serves 
as a carrier for the heat. When steam performed this 
office it appeared that the P F-diagram was compounded 
of two horizontal rectilinear isotherms and two hyperbolic 
adiabatics, the area enclosed being quite considerable. In- 
deed, the actual steam-engine, whose success has always 
been founded chiefly upon its large amount of power de- 
veloped per unit-weight and unit-volume of working-sub- 
stance, works a close approach to the Carnot cycle. But 
if air or any other permanent gas be the working-substance, 
the isotherms and adiabatics are both hyperbolic in form 
and are very nearly parallel. The P /^-diagram enclosed 
between any intersecting four of them, as shown in Fig. 18, 
is very long and narrow in proportion to its area. The 
range of both pressure and volume per unit of work devel- 



130 



THE THERMODYNAMICS OF HEAT-ENGINES 



oped is enormous. Entirely aside from the obstacles of 
bulk and heating-surface which have opposed the success 
of all air-engines, this additional one of wide range of 
pressure and volume per unit of work has always prohibited 
all hope of seeing a successful Carnot-cycle air-engine. 




Fig. 18. 



The obstacle to efficiency in the steam-engine is not the 
poor form of its cycle, but is the narrow practicable range 
in temperature. It already begins to appear that the ob- 
stacle to efficiency in the use of the permanent gases as 
working-substances in heat-engines is not any limitation in 
temperature-range, but in the poor form of their practicable 
cycles. This contrast will appear more strongly in the 
following chapters. 



CHAPTER VI 

THE GAS-ENGINE CYCLES 

The history of the public distribution of artificial illumi- 
nating-gas opened with the century which has recently 
come to its close. Murdoch, Watt's famous assistant, had 
an active hand in the matter, and the shops of Boulton & 
Watt were amongst the first buildings upon which the 
experiment was tried. The presence of the new fuel very 
quickly led to attempts at its utilization for the develop- 
ment of power. Patents for gas-engines begin to appear 
upon the records before 1825. The second quarter of the 
century is thickly sprinkled with them. It was not until 
i860, however, that an engine was produced which was 
sufficiently successful in a mechanical sense to attain a 
commercial basis, and for a long time thereafter the price 
of gas and the fuel-consumption of the engines both re- 
mained so high that very little progress was made in their 
adoption as a standard prime mover, even in the small 
sizes to which they were then restricted. The present 
position of the internal-combustion engine is due more to 
Dowson's invention of a cheap power-gas generator, to the 
natural-gas supply of the United States, to the utilization 
of the waste-gases from blast-furnaces for power-develop- 
ment, and to the wide and cheap supply of the lighter 
petroleum distillates, than it is to the development either 
of illuminating-gas or of the engine itself. 

The Lenoir Cycle. — The first gas-engine sufficiently 
successful to command a market was the Lenoir, brought 

131 



132 THE THERMODYNAMICS OF HEAT-ENGINES 

out by a Frenchman of that name in i860. The Lenoir 
engine much more closely resembled a single-cylinder 
horizontal steam-engine of the throttle-governed type than 
has any gas-engine since then. It was double-acting and 
controlled by a fly-ball governor. There the similarity 
ended, however. Its cylinder was water-jacketed, the 
valves were quite different, and the boiler was absent. 

The cycle of processes in the Lenoir engine was as 
follows : — 

(1) From dead-centre the fly-wheel drew the piston out, 
sucking in as it did so a charge of gas and atmospheric air 
in the correct proportion to form an explosive mixture ; 

(2) Somewhat before half-stroke was reached the inlet- 
valve was closed and the charge was ignited, producing an 
explosion which elevated the pressure in the cylinder con- 
siderably above atmospheric ; 

(3) The remainder of the stroke was devoted to expand- 
ing this pressure to atmospheric, or nearly so ; and 

(4) The return-stroke exhausted the burnt gases. Thus 
there was produced an impulse upon each side of the 
piston at each stroke, or two for each revolution within a 
single cylinder. 

The objections which led to the abandonment of the 
Lenoir engine were (a) the great consumption of fuel, 
(b) the violent overheating of parts, especially the piston, 
and their consequent rapid deterioration, and (c) the slight 
power developed. 

Hugon was the only other builder of a true Lenoir 
engine, but the thermodynamic cycle survived in the free- 
piston engines of Barsanti & Matteucci and of Otto & 
Langen, who tried to improve Lenoir's fuel-consumption 
on the theory that it was the rapid loss of heat to the 
jacketed walls immediately after explosion which was the 
source of trouble. They therefore built vertical engines 



THE GAS-ENGINE CYCLES 1 33 

each with a long cylinder in which a heavy piston was 
driven rapidly upward by the explosion; but the piston 
instead of being attached to a crank and limited in speed 
by its motion was free to rise to the top of the cylinder at 
the maximum velocity possible. It carried with it a rack 
meshing with a pinion which rotated loosely on the over- 
head fly-wheel shaft. The shaft normally revolved in the 
direction opposite to that of the pinion during the upward 
stroke of the piston. When the force of the explosion 
and of inertia had been overcome by gravity, the piston 
descended, aided by the partial vacuum formed beneath it 
by the cooling of the spent charge, and the motion in this 
direction engaged the pinion with the shaft by a pawl-and- 
ratchet device. It was on this return-stroke that power 
was imparted to the shaft. 

These free-piston engines were better than the Lenoir 
as to fuel-consumption ; they also remained cooler and 
worked more reliably. On the other hand they were fear- 
fully noisy, too noisy for toleration, and in their turn they 
have passed away. 

The only surviving representative of the Lenoir cycle 
is the Bischoff, a very small vertical engine whose sole 
recommendation is simplicity. 

Problem 27. What are the conditions of operation and 
the efficiency of a Lenoir engine using at each stroke one 
cubic foot of an explosive mixture consisting of nine-tenths 
air and one-tenth gas of a calorific power of 650 B.t.u. per 
cubic foot, the atmospheric pressure and temperature being 
14.5 pounds and 59 F. respectively ? 

The first operation, that of drawing in the charge, is 
shown at AB, Fig. 22. Since this is not a thermodynamic 
process, but merely a change of location, it does not appear 
in the iVT^-diagram. Hence, V B — 1.00, P B = 14.5 and 
T B = 520. 



134 



THE THERMODYNAMICS OF HEAT-ENGINES 



As to the entropy, since we are no longer relying upon 
the steam-tables, we need no longer adhere to their arbitrary 
zero of 32 F. It will be preferable to assume an arbitrary 
zero-axis for entropy which passes through the initial point 
of this problem, B. Hence, N B = o. 

Although the piston is really in quite rapid motion at B, 
and although an appreciable amount of time is required 




Fig. 19 



for combustion to take place, yet theoretically the explosion 
is instantaneous and the piston accomplishes no motion 
while it is taking place. The next process therefore con- 
sists of the development of heat within the working-sub- 
stance at constant volume. This process may be regarded 
as an addition of heat from without, for the heat is just as 
truly added to the substance when coming from its own 



THE GAS-ENGINE CYCLES 



135 



chemical alteration as when coming from outside. The 
process is therefore properly represented by the curve 
BC, Fig. 19, whose equation is 

N-N B =S r log e -^, 

1 B 

and by the line BC of Fig. 20, whose equation is 
V= a constant = V B . 




Fig. 20 



Moreover, the volume being constant, 

Pb T b 

The available heat per unit of mass, measured by the area 
OBCd, Fig. 19, is S r (T c — T B ). The mass present, M, 
may be found from the equation 

PV= CT-M, or 14.5 x 144 x 1 = 53.18 x 520 x M, 



136 THE THERMODYNAMICS OF HEAT-ENGINES 

whence M= 0.0758. * But since one-tenth of the cubic foot 
consists of gas having a density which may be assumed to 
be 0.6, the true mass is 0.96 x 0.0758 = 0.0727. The 
available heat is 65 B.t.u. From these equations are 
derived 

T C -T B = 5315, 

^=5835, 

N c = 0.02044, 

and D c = 162.7. 

The process CD must be an adiabatic, unless artificially 
distorted therefrom. Therefore in Fig. 19 the equation of 
CD is 

N= 0.02044, 
and in Fig. 20 

P V l - m r= a constant. 

From Equation 35 is derived the value of T D as 2915. 

In Fig. 19 the areas beneath the curves give the heat 
handled, and in either diagram the enclosed area gives the 
work done. Q 1 is known to be 65 B.t.u. ; Q 2 , or the area 
OBDd, = MS P (T D - T B ) = 41.3 B.t.u. Hence Q w = Q 1 
— <2 2 = 18.7 B.t.u. and the efficiency is 28.8%. 

It is needless to say that no such temperatures or effi- 
ciencies were ever attained in the actual engine ; the walls 
absorbed too much heat. 

The Beau de Rochas Cycle and the Otto Engine. — At 
about the time when Otto & Langen were experimenting 
with their free-piston engine, M. Beau de Rochas enun- 
ciated the principles upon which a gas-engine might be 

1 In the majority of cases, however, the accuracy of conclusions will not be 
affected if the assumption be made, for convenience, that M— unity, the 
quantity of heat being calculated per unit of mass instead of per unit of 
volume. 



THE GAS-ENGINE CYCLES 1 37 

operated successfully and economically. A few years 
later, in 1876, Dr. Otto brought out his "Silent" engine 
embodying them. The name was designed to signify that 
the objectionable noise of his earlier engine was no longer 
present. 

This cycle, usually but erroneously referred to as the 
Otto cycle, consisted of the following processes : — 

(1) The drawing into the cylinder throughout an entire 
stroke of a charge of explosive mixture at atmospheric 
pressure. 

(2) The return of the piston under the influence of the 
fly-wheel, compressing this charge into the clearance space, 
which is much larger than in steam-engines. 

(3) The ignition and explosion of the charge while the 
piston is at rest on dead-centre, the process being now 
truly under constant volume. 

(4) The expansion of the heated gases throughout the 
entire stroke. 

(5) The exhaust of the burnt gases during the back- 
stroke. 

The Otto Silent engine had a great and immediate suc- 
cess. The bulk of the gas-engines built since 1876 have 
closely followed the principles of mechanical design which 
it embodied, as well as its thermodynamic cycle. It has 
nearly always been built in single-cylinder, single-acting 
form, in order to keep one side of the piston open to the 
air and thus avoid the overheating of the Lenoir. In con- 
sequence, it gets but one impulse every four strokes, or 
every two complete revolutions. Although at first but 
slightly better in fuel-consumption than the free-piston 
engine, owing to too little compression, it was quiet, power- 
ful, and reliable in operation, and in every way an improve- 
ment over what had preceded. It is now being built in 
two-, three-, and four-cylinder form, in order to multiply 



138 THE THERMODYNAMICS OF HEAT-ENGINES 

impulses. One or two builders have produced double- 
acting engines. For many years its size was limited to an 
average of less than five horse-power, partly by reason of 
the cost of fuel and the consequently limited field of in- 
dustry in which it was profitable, and partly by reason of 
mechanical obstacles which multiplied rapidly with increase 
in size. To-day the natural-gas supply of the Mississippi 
Valley, and the use of the waste gases from the blast-fur- 
nace for driving gas-engines in Continental Europe have 
raised the practicable limit in size of a single engine to 
4000 horse-power, with no reason apparent why further 
progress in capacity may not be made. 

The Clerk-cycle or ' ' Two-cycle ' ' Gas-engine. — The ob- 
vious mechanical objections arising from the limitation of 
the standard form of the Otto gas-engine to one impulse 
for each two revolutions led to its modification by Mr. Clerk 
so as to obtain an impulse once each revolution. To accom- 
plish this a smaller preliminary or metering cylinder was 
added to the engine. Its office was to draw in the explosive 
mixture of atmospheric air and gas to the amount required. 
Its piston was so timed in its motion that it had partially 
accomplished its return-stroke, slightly compressing the 
contained charge, when the main piston arrived at dead- 
centre after the completion of its expansion-stroke. Com- 
munication between the two cylinders was then opened, 
allowing the slightly compressed fresh charge to enter the 
main cylinder, driving before it the burnt gases out the ex- 
haust-valve. At the moment when the fresh charge had 
practically displaced the burnt one both exhaust- and admis- 
sion-valves were closed, and the return of the main piston 
prepared the fresh charge for ignition without the media- 
tion of a pair of exhaust- and suction-strokes. 

As built now, chiefly for marine and locomobile work, or 
in the smaller sizes for stationary work, the "two-cycle" 



THE GAS-ENGINE CYCLES 1 39 

engine is nearly always a single-acting vertical engine, with 
the connecting-rod and crank working in an enclosed crank- 
case. This crank-case is employed for the preliminary 
metering of the charge, instead of a separate cylinder. As 
the piston rises, the fresh charge is drawn into the crank- 
case, below the piston. At upper dead-centre the admis- 
sion-valve is closed and the downward stroke slightly 
compresses the charge. At lower dead-centre communi- 
cation opens between the crank-case and the cylinder, and 
the fresh charge expands into the cylinder, displacing the 
burnt charge. The upward stroke of the piston then per- 
forms the usual compression of the charge preliminary to 
ignition. 

The builders and buyers of gas-engines draw considerable 
distinction between the so-called "Otto-cycle" and "two- 
cycle " engines. It is true that, although much skill has 
been expended in markedly improving the two-cycle type, 
yet there will always be an imperfection in its action and 
efficiency due to the impossibility of displacing instantane- 
ously the burnt charge by the fresh one without either 
losing some of the latter through the exhaust-valve or 
retaining some of the former within the cylinder. 1 It is 
therefore proper to draw a distinction between the two. 
Yet it is obvious that the difference is mechanical, not 
thermodynamic, in character. The preliminary " compres- 
sion " is merely a means of displacing the burnt charge. 
Since the exhaust-valve is wide open at the instant that 

1 In the large Korting engines, in the region of 1000 horse-power each, 
which operate upon this cycle, this difficulty is obviated by having two dis- 
placer-cylinders. One of them pumps fresh air through the main cylinder 
immediately after exhaust, cleansing or " scavenging " the cylinder of all burnt 
gases. The other follows with the fresh charge of explosive mixture, small 
enough in volume to ensure that none of it shall pass out through the exhaust- 
port, and rich enough to stand dilution with the remnant of fresh air remain- 
ing after the scavenging. 



140 



THE THERMODYNAMICS OF HEAT-ENGINES 



the charge enters the main cylinder, the pressure must be 
atmospheric. Whatever preliminary compression there 
was has been lost. Thermodynamically, the Otto-cycle 
and the two-cycle engines operate upon exactly the same 
cycle. 

Problem 28. What are the conditions of operation and 
the efficiency of an Otto engine working under the same 

T 




Fig. 21 



premises as the Lenoir of Problem 27, except that the 
charge is compressed to an absolute pressure of 75 pounds 
per square inch before explosion ? 

The cycle is depicted in Figs. 21 and 22. AB is the 
suction of the charge, not appearing in the AT-diagram. 
BC is the adiabatic compression, CD is the explosion, DE 
the adiabatic expansion, and EB the exhaust at constant 
volume at the end of the stroke. The starting-point of 
the calculation of the cycle may now be taken as based 



THE GAS-ENGINE CYCLES 



141 



upon the fact that V AB = 1 cubic foot. If P is to be 75 
pounds per square inch, then 



(VA 1Am _Po_ 

\Vj Pn 



75 



VJ P B 14.5 

V B =1+V C . Therefore, 

V Q —V A = 0.452 and V B = 1.452. 

T c =837- N B = N c =0. 

To the mass of the 1 cubic foot of explosive mixture of 
Problem 27 must now be added the 0.452 cubic foot of burnt 




b N 



Fig. 22 



gases in the clearance-space, or M= 0.1057. Proceeding 
as before, T D = 4498, N D = 0.02972, and P D — 403. The 
expansion DE cannot be carried down to atmospheric 
pressure, because the engine's cycle limits the final volume 
after expansion to the initial volume before compression. 1 



1 See page 287, Part II, et seq. 



142 THE THERMODYNAMICS OF HEAT-ENGINES 

P P 

Hence, V E = V B = 1.452. -^ = -=£■ P E = 78, and 
^=2792. Fe \ £ 

The two isomorphic curves CD and BE of Fig. 21 have 
the same equation, except that the coefficients of specific 
heat differ ; the curves DE and CB of Fig. 22 bear a sim- 
ilar relation. Q x is the area OCDe (Fig. 21), and Q 2 is 
the area OBEe. But geometrical inspection will prove 
that these areas must bear the same proportion to one an- 
other as T c to T B , or as P B to T E } Therefore Q w = Q 1 — Q 2 
must bear the same proportion to Q x or Q 2 as T c — T B does 
to T c or 7^, respectively; 

_„ Qw Pc ~ 1 u Pb ~ Pe /,o\ 

or 7T- = t = t ' ^> 

from which the efficiency of the cycle is found to be 37.9^. 
It is obvious from this that it is the vertical separation 
of the curves CD and BE of Fig. 21, or the proportion of 
the total available fall in temperature which is utilized, 
which determines both the efficiency and the power of the 
engine. This explains why the early Otto engines, with 
their slight degree of compression, were so slightly more 
economical than the Lenoir. It also explains why the 
chief line of progress in gas-engine design between the 
years 1875 and 1895 has consisted in increasing the degree 
of compression. Progress in this direction has now almost 

1 The area beneath CD, Fig. 21, is equal to an integration of the expression 
TdN. The area beneath EF is measured by the same expression. For any 
given value of dN or of N, however, the value of these two expressions will be 
the proportional to the values of T. Since the equation of CD is N— Nq 

= S loge — and that of EB is N — A^o = S v log e — , for any given value of 

Tc T B 

A^the corresponding values of T for the two curves must be proportional to 

— -• Hence, the areas beneath CD and EF, between any given pair of ordi- 

Tb rp 

nates, are proportional to ~^r' 



THE GAS-ENGINE CYCLES 1 43 

ceased, (1) because of the rapidly increasing mechanical 
objections due to the development of such high momentary- 
pressures at the point D without proportionate increase in 
work done : the engine must be strong enough, in castings 
and bearings, to stand this maximum pressure, and yet, be- 
cause the pressure is so evanescent, the engine derives little 
good from it ; (2) because of the danger of preignition when 
the degree of compression is high, particularly in oil-engines. 
The Joule Cycle. — -The Joule cycle was proposed by him 
who discovered the mechanical equivalent of heat. It has 
found its embodiment in several designs of engine, none 
of which have survived the test of competition with the 
Otto type. Some of them have been gas-engines, as have 
the Gardie and the Bergher, some oil-engines, as the 
Priestman, and some almost purely hot-air engines, as 
the Cayley-Buckett. The Joule type is nevertheless of 
the greatest importance because, while no one has ever 
yet put it into profitable practice, it offers possibilities of 
superiority over the Otto which have attracted to it a long 
line of inventors. Their failures may in each case be 
easily explained as inevitable under the mistaken line of 
design which they adopted. It is the personal opinion 
of the author that the large gas-engine of the future will 
operate on no other than the Joule cycle. His reasons, 
however, are not solely, nor even largely, based upon those 
thermodynamic points of superiority which are developed 
in the following pages. They are based chiefly upon 
purely mechanical considerations which lie outside the 
limits of the present discussion. It is worthy of note, 
however, that the purely thermodynamic aspect of the 
question shows that under any chosen limits of tempera- 
ture and pressure, the theoretic efficiency of the Joule 
cycle is not only greater than that of the Beau de Rochas 
cycle, but it is greater than that of any other cycles known 



144 



THE THERMODYNAMICS OF HEAT-ENGINES 



to have been proposed or put into practice except the 
Carnot, the Stirling, and the Ericsson. Of these three, all 
are known by long trial to be impracticable. 

The essential features of an engine operating on this 
cycle are shown in Fig. 23. They comprise two cylinders, 
C and W, the latter larger than the other, the two being 
combined in any way so that power may be transmitted 
from one to the other ; a source of heat at a higher tem- 
perature 7\ ; a refrigerator at a lower one T 2 . Supposing 





Scarce of 










/*" Heat "N. 










T, - 










» - 1 












H 










w 






Hn 






c 


» 


1 — 


=\ <y> — 




Source o 


1 










v. r. j 










Fig. 23 

the entire system to be filled with the chosen permanent-gas 
working-substance, air for example, the operation is as 
follows, the processes being traced in the diagrams of 
Figs. 24 and 25 : — 

(1) There is drawn into cylinder C a charge of working- 
substance. This is shown by AB of Fig. 25 ; it does not 
appear in Fig. 24. 

(2) The adiabatic compression of the charge to the max- 
imum working-pressure of the engine. This is shown by 
BC in both diagrams. 



THE GAS-ENGINE CYCLES 1 45 

(3) The addition of heat under constant pressure ; CD 
in both diagrams. 

(This process really involves the triple process of (a) the 
expulsion of the compressed charge from the cylinder into 
the chamber T u shown by Cc of Fig. 25 ; (b) its increase 
of volume while there ; (c) the taking of the increased vol- 
ume into the cylinder W, which is shown by cD of Fig. 25. 
But as this combination consists entirely of changes of 
locality, it need not be expressed upon the thermodynamic 
diagrams.) 

(4) The adiabatic expansion of the heated gases to 
atmospheric pressure. This is shown by DE in both 
diagrams. 

(5) Their exhaust at constant (atmospheric) pressure. 
This is shown by EB in both diagrams. 

It is immediately obvious that the process of heating 
the gases in the chamber T x may be done either by the 
transmission of heat to them through the surrounding 
walls or by their own internal combustion. If the latter 
be the case, the combustion must take on the character- 
istics of a flame, increasing in volume under constant press- 
ure, rather than those of 'an explosion, which implies 
an increase of pressure under constant volume. In the 
Cayley-Buckett engine this chamber was a furnace enclos- 
ing an anthracite fire ; the furnace was ordinary in every 
respect except that it was capable of withstanding internal 
pressure. In the Priestman oil-engine the working-sub- 
stance was a mixture of oil and air ; this chamber T x was a 
little combustion-chamber fit to contain an oil-flame and 
separated from the cylinder C by a shield of wire gauze to 
prevent the flame from travelling back into the compressed 
charge which was being driven out of C. In the Gardie 
and Bergher engines the combustion takes place in the 
cylinder W as the compressed charge enters. In all 



146 



THE THERMODYNAMICS OF HEAT-ENGINES 



except the first the combustion is intermittent, ignition 
being provided every time that the cycle reaches the 
point C and the flame being extinguished at the point D ; 
in the first the flame burns more or less continuously, the 
chamber T x being sufficiently large to store the extra 
energy evolved during the time when the cylinder W is 
not taking its charge. 




Fig. 24 



Problem 29. What are the conditions of operation 
and the efficiency of a Joule engine working under the 
same premises as the engines of the two preceding 
problems, except that the charge is compressed to a 
pressure of 365 pounds per square inch absolute before 
ignition ? 

Here the original cubic foot of explosive mixture appears 



THE GAS-ENGINE CYCLES 



147 



as the volume AB. Calculating the values for the several 
points as before, it appears that V c = o. ion, T c = 1323, 
Vj) = 0-3995, T D = 5095, N D = 0.02438, T E = 2002, and 
V E = 3.844. As in the case of the Otto cycle, the curves 
CD and BE of Fig. 24 have the same equation and differ 
only in the point of origin ; therefore the area beneath 
each curve is proportional to the ordinates at any ab- 
scissa, or Q w : <2i as area BCDE : area OCDe — BC : OC = 
803 : 1323 = 60.7% efficiency. 




Fig. 25 



Although it is obvious from these figures that the effi- 
ciency of the Joule cycle is much superior to either the 
Lenoir or the Otto under the conditions chosen, it is not 
equally clear that it would be under any conditions. The 
diagrams are not drawn to the same scale nor, even if so, 
could any set of conditions be chosen which would be 
equally fair for all three. But an excellent general com- 
parison may be had by adding to Figs. 24 and 25 the 



I48 THE THERMODYNAMICS OF HEAT-ENGINES 

constant-volume lines CD and BE' . Then between the 
one pair of adiabatics, BC and DE, are shown ; 

(1) A Lenoir cycle, BE'E; 

(2) An Otto cycle, BODE' ; and 

(3) A Joule cycle, BCDE. The comparison is not fair 
to the Lenoir cycle, because the efficiency of the latter 
increases with its width ; hence, in the Lenoir, the richer 
the gas or the oil used, the better will be the efficiency. 
But with both the Otto and the Joule cycles the efficiency 
is independent of their width in the TVT-diagram, being 
solely dependent upon their height, or upon the degree of 
compression. Therefore, their limitation between a given 
pair of adiabatics is equally fair to both. Since the Lenoir 
cycle is out of the race in any event, the injustice done 
to it may be overlooked. 

It is further obvious that the territories of the Lenoir 
and of the Otto cycles are mutually safe from trespass. 
The Lenoir type cannot get into the territory of the Otto, 
to the left or above BE ; for it may not embody initial 
compression. The Otto cannot cross the same line in the 
opposite direction, into the territory of the Lenoir, without 
final expansion to a volume greater than its initial one, 
which is impossible. 

The avoidance of this inherent limitation of the Otto 
cycle has been sought, in a single-cylinder type, by Mr. 
Atkinson in his "Differential" and "Cycle" engines, and 
by various designers in the multiple-cylinder type by add- 
ing larger low-pressure cylinders in which the expansion 
E'E might be performed; but all have failed, the first for 
mechanical and the second for thermal reasons. 

It is also sought in some of the largest and best engines 
of the present day, without added mechanism, by restrict- 
ing the volume taken in at atmospheric pressure to less 
than the full cylinder-volume. This also fails to attain 



THE GAS-ENGINE CYCLES 1 49 

the desired end, for reasons which are explained on 
page 287. 

So, with this limitation accepted as inevitable, and 
remembering that the chief limitations in constructive 
practicability are pressure and temperature, the broad con- 
clusion may be safely drawn that tender the same maxi- 
mum limits of pressure and temperature, the Joule cycle has 
inevitably the greatest efficiency of the tliree, its area not only 
completely covering the Otto and the bulk of the Lenoir, but 
having some territory of its own unattainable by eitlier of 
the others. 

The universal failure of the Joule-cycle engines in the 
past can be explained, in the face of this statement, by the 
fact that in no case was a working-pressure adopted which 
would permit any possibility of success. Always arbitrarily 
limited by their designers to about 100 pounds as their 
maximum pressure, the Joule-cycle engines hitherto con- 
structed could have no hope of successful competition with 
the Otto-cycle engines, when the latter regularly attained 
pressures of from 300 to 400 pounds as maxima. 

The Diesel Cycle. — In the year 1897 Mr. Rudolph Diesel 
announced to the world the successful operation of an 
engine operating under a cycle which he had previously 
advocated from purely thermodynamic considerations. 1 In 
the argument which led him to his result Mr. Diesel started 
in search of the nearest possible approach to a Carnot cycle. 
Compression was made as purely adiabatic as possible, 
instead of as nearly isothermal as possible, as had usually 
been done in the embodiment of previous cycles. High 
efficiency being the main object in view, a wide range of 
temperature was needed, and this called for a similar range 
in pressure. The working pressure was therefore fixed as 
500 pounds per square inch. As adiabatic compression to 
lt( A Rational Heat-motor," by Rudolph Diesel. 



150 THE THERMODYNAMICS OF HEAT-ENGINES 

this pressure resulted in a temperature above that sufficient 
for the ignition of the fuel finally chosen, which was oil, 
igniting devices were omitted. The oil was simply injected 
into the cylinder when compression was complete, into an 
atmosphere above the temperature of ignition and in which, 
therefore, it could not help burning. The first result was 
a still further increase in temperature ; but as a Carnot 
cycle demands isothermal addition of heat, Mr. Diesel 
invented his famous isothermal combustion, in which merely 
enough fuel was admitted to maintain a constant tem- 
perature as the gases expanded. This process replaced 
the ordinary temperature-raising combustion soon after the 
stroke commenced, and continued until the point in the 
stroke where the governor dictated a cut-off of fuel-supply. 
After this adiabatic expansion set in. 

The general arrangement of the engine was similar to 
that of a vertical Otto-type engine, except that the cylinder 
was longer and narrower, and there was added an air-pump 
for creating a pressure on the surface of the oil sufficient 
to drive it into the compressed air within the cylinder when 
the valve opened. One impulse was received every two 
complete revolutions, as in the Otto engine. 

The Diesel cycle is portrayed in Figs. 24 and 25 by the 
area BCFGE' B, though a much higher pressure-limit than 
that drawn must be imagined in comparing it with the other 
cycles. The compression to 500 pounds is shown by BC, the 
primary combustion by CF, the isothermal combustion by 
FG, the adiabatic expansion by GE' , and the exhaust at 
lower dead-centre by E'B. It is obvious that the cycle is 
much nearer to a Joule than to a rectangular Carnot. It 
is further obvious that Mr. Diesel, in remembering the 
statement that the Carnot cycle was the one of maximum 
efficiency, forgot under what limitation that statement was 
made : " Under any given temperature-limits." For if 



THE GAS-ENGINE CYCLES 151 

some other cycle depart from the Carnot rectangle but at 
the same time attain a greater temperature-range, it may 
have an efficiency greater than that to which the original 
Carnot cycle was restricted. Thus, here, the isothermal 
FG, which would impart greater efficiency than the iso- 
morphic CD, if the upper temperature -limit were FG, plainly 
decreases the efficiency when compared with that attained 
by admitting the heat beneath FG along the isomorphic 
FD, to some point near D, before expansion sets in. 

From these considerations it may be concluded that the 
maximum efficiency available under equal pressure-range 
is always greater in the Joule than in the Diesel cycle. 
Under equal temperature-limits either may be the greater ; 
but to attain equal temperature-limits the Diesel cycle de- 
mands an egregiously superior maximum pressure. 



CHAPTER VII 

THE HOT-AIR ENGINES 

The history of the hot-air engines, if the efforts of Hiero 
of Alexandria, who worked in the second century before 
the Christian era, and of a few unsuccessful inventors be 
excepted, is comprised between the years 1827 and 1854. 
On the former date Robert Stirling took out his patent for 
a regenerative air-engine ; on the latter date Ericsson's hot- 
air propelled ship, with its enormous engines, sank in New 
York harbor, never to rise again into a world of " caloric " 
engines. During this period the energy of many of the 
best minds of the mechanical world were devoted to its 
perfection. It passed from the status of a toy to a prime 
mover of hundreds of horse-power, and back again. It ex- 
perienced all the glamour and romance to be had from 
youthful promise of the most fascinating sort. It grew to 
manhood under the guidance of master minds. It was 
tried and found wanting. It is now relegated almost solely 
to the task of supplying occasional country-residences and 
small hotels with water. The question of its worth for any 
larger task than this has been effectually sealed by one of 
the most valuable failures on the part of one of the ablest 
engineers that the world has ever seen, — that of Ericsson, 
who, after twenty years of effort, failed to produce a ma- 
chine which could compete with the steam-engine in the 
world's field. 

From the very start the hot-air engine offered an irresisti- 
ble fascination for the inventor, because even the crudest 
attempts with it always resulted in an efficiency far superior 

152 



THE HOT-AIR ENGINES I 53 

to that of the steam-engine. This is readily explained as 
being due to the far greater temperature-range possible 
when working with a perfect gas than with steam, where 
the attempt to get high temperatures is always met by a 
need of impossibly high pressures. This contrast was of 
course more marked in those days, when 30 to 50 pounds per 
square inch was regarded as the highest steam-pressure 
practicable in stationary engines. Its final failure was due 
to phenomena which manifested themselves from the start 
and which are now known to be ineradicable. These 
phenomena are : 

(1) The rapid deterioration of metallic surfaces when 
used to transmit high-temperature heat from an atmos- 
phere of carbonic acid to a hot dry gas, such as air ; 

(2) The large amount of such surface required to transmit 
a given amount of heat when compared with like transmis- 
sion to water or to saturated steam ; and 

(3) The great bulk of the weight of air requisite for 
handling a proper amount of energy. 

The first two obstacles have been obviated in the natural 
offspring of the hot-air engine, the internal-combustion gas- 
and oil-engines. Incidentally the last has also been partly 
eliminated in the same way ; though it is imaginable that 
the application to the hot-air engine of as great a working- 
pressure as is now common in the gas-engine might have 
been very beneficial to it in respect to bulk. 

The Stirling Air-engine. — The utter impossibility of a 
practicable Carnot-cycle air-engine was pointed out at the 
end of Chapter V. The attainment of any practicable ef- 
ficiency by means of any other cycle, with air as a working- 
substance, became only possible with the invention of the 
regenerator, by Stirling, in 1827 as a substitute for adiabatic 
compression and expansion for the transfer of the air from 
one temperature-level to another. The regenerator consists 



154 THE THERMODYNAMICS OF HEAT-ENGINES 

of a chamber containing a mass of heat-absorbing material 
arranged so as to present as large a surface as possible to 
a current of air passed through it. In practice this mate- 
rial was usually iron wire, arranged in a close network. 
Through this network the air was passed back and forth, 
giving up to the wire its heat as it entered from the hotter 
end and picking it up again as it passed back again from 
the colder refrigerative chamber of the engine. The only 
forms in which the regenerator survives in modern industry 
are the hot-blast " stoves " of the blast-furnace, and the 
air-heating surfaces of the regenerative gas- and oil-stoves 
and lamps. With all of the work in this line the name of 
Siemens is chiefly and actively connected. 

The arrangement of Stirling's regenerative engine is 
shown in Fig. 26, in which the regenerator occupies the 
annular space RR about the larger cylinder. When in 
operation the temperature of this generator at the lower 
end is 7^ and at the upper end T 2 , being graded continu- 
ously from one to the other. Constructively, the engine 
consists of a crank-shaft and fly-wheel to which are coupled 
by connecting-rods the large " displacer piston" D and the 
"working-piston " W, but in such a way that their strokes 
take place alternately. (Theoretically, the stroke of each 
should be complete before that of the other begins. In 
practice, each stroke laps over the other somewhat ; but in 
analyzing the engine's action the first will be assumed to 
be the case.) At the lower end of the cylinder D is placed 
the fire, in the furnace F, usually maintaining the iron at a 
temperature approaching a visible red, and at the upper 
end is the coil of condenser-pipes CC, filled with a current 
of cold water. The action is traced mathematically in the 
AT and PV diagrams of Figs. 27 and 28. 

It were best noted at the start that the volume of air 
within the engine depends solely upon the position of the 



THE HOT-AIR ENGINES 



155 



piston W. The motion of the piston D, which fits loosely 
in its cylinder, meets with no possible resistance and causes 
no change of volume, because the two ends of its cylinder 
are always open to one another. 

The engine is shown in a position represented as midway 
between the points A and B in the diagrams, containing a 




Fig. 26 

quantity of air at approximately atmospheric pressure and 
minimum temperature. The air had been at atmospheric 
pressure at the beginning of Ws downward stroke, at the 
point A. It has now been somewhat compressed, by the 
energy of the fly-wheel, isothermally because in contact 
with the cold pipes CC. This process is shown by the line 
AB, in both diagrams. At the end of Ws stroke the 



1 5 6 



THE THERMODYNAMICS OF HEAT-ENGINES 



volume is a minimum, and D now moves upward, displacing 
the cold air from the space CC through the regenerator RR 
into the lower end of the cylinder. (The by-passing of the 
air around the regenerator through the ports leading to W 
is properly prevented, sometimes by valves, sometimes by 
the small size of W as compared with R and D, and some- 
times by making use of only one port.) This raises the 




Fig. 27 



air, at constant volume, from the pressure at B to that at C 
by reason of the change of temperature from T 2 to T v 
This is shown by BC in both diagrams. 

While D is now quite at its upper dead-centre, W moves 
up under the influence of the increased pressure and makes 
its working-stroke, the air expanding isothermally instead of 
falling in temperature adiabatically, because in contact with 
the fire-surface at the temperature T' . This is shown by 
the line CD in both diagrams. 



THE HOT-AIR ENGINES 



157 



Then, while W is quiet at its dead-centre, D comes down, 
displacing the air again to the top of the cylinder through 
the regenerator, cooling the air under constant volume from 
7\ to T 2 and reducing the pressure from P D to P A . This 
is shown by the line DA in both diagrams. The cycle is 
then complete. 

Thus the displacer-piston D, with the aid of the regen- 
erator, changes the pressure; the working-piston W, with 




Fig. 28 



the aid of fire and condenser, changes the volume, isother- 
mally. This cycle may be summarized as follows : 

I. D at bottom ; W moving down ; line AB in diagrams ; 
negative work done (area under AB, Fig. 27). 
II. Wat bottom ; D moving up ; line BC in diagrams. 

III. D at top ; W moving up (working-stroke); line CD. 

IV. W at top ; D moving down ; line DA in diagrams. 



158 THE THERMODYNAMICS OF HEAT-ENGINES 

Problem 30. What are the conditions of operation and 
the efficiency of a Stirling engine operating with one cubic 
foot of atmospheric air, the temperature of the fire-surface 
being 1539 F. and the volume of the rest of the engine 
being four times that of the cylinder W? 

This gives the volume at A = 1.00 cubic foot, that at 
B = o.8o cubic foot. The temperature at A and B is 520 F. 
absolute, and that at C and D is 2000 absolute. The press- 
ure at A is 14.5 pounds per square inch. 

/4i5 being an isothermal, P B = — — X 14.50= 18.12 pounds 

per square inch. In the heating at constant volume the 
pressures must change proportionally with the tempera- 
tures; therefore P c = x 18.12 = 69.25 pounds. CD 

being an isothermal again, P D = — — x 69.25 = 5 5.40 pounds. 

The mass involved being 0.0758 pound and the heating 
being at constant volume, ^=0.001301. The heat ab- 
stracted along AB must be equal to the (negative) external 
work done, or 

Q* = —'PaV a log, r = ^^ ■ \og e — 
' ^ 2 778 A A 6e 778 Be V B 

53.18x0.0758x520 . 1. 00 

= — ~ 3 — x log e —5- = 0.601 1 B.t.u. 

778 &e o.8o 

The entropy absorbed along AB must be this divided by 
520, or 0.001156. Substituting T D for T A in the equation 
for ft gives Qi = 23I2BtM ; y 

Q*=Qi-Q*= 171 1 B.tu., 

As to efficiency, it might seem from the diagram that the 
heat supplied by the fire should be the area beneath BCD, 



THE HOT-AIR ENGINES 1 59 

Fig. 27, and that that absorbed by the refrigerator should 
be the area beneath DAB. But of these quantities the 
heat beneath BC enters the air, not from the fire, but from 
the regenerator, where it was stored up by the DA -process 
of the preceding cycle. Thus, while it might seem from 
the form of the cycle that the heat fell into the BC-\ine 
from the temperature T v and out of the DA-line into 7" 2 , 
this is not the case, as will be evident if two succeeding 
cycles, such as A BCD and aADd, are considered. Each 
atom of entropy, such as that at t, which reaches the AD- 
line of the second cycle has done so by first falling from t x 
to /, doing work in the first cycle ; it is deposited, without 
free fall, in the regenerator at this level. It is afterward 
picked up by the ^!Z>-process of the second cycle, in which 
it falls from t to t 2 , again doing work. So the heat sup- 
plied, Q v is only that beneath the line CD, and Q 2 is only 
that beneath AB. As the entropies of these two processes 

T 
are equal, ^l = — 1, and the efficiency is that of a Carnot 

cycle, or, in this case, 74.0%. 

Engines operating upon this cycle were built as large as 
50 to 75 horse-power before their bulk and the lack of 
durability of the heating-surfaces threw them out of com- 
petition with the steam-engine. 

The Ericsson Hot-air Engine. — The Ericsson engine 
differed from the Stirling constructively in that, instead of 
the working-piston also performing the compression, while 
the displacer-piston meets with no resistance, both pistons 
are under pressure, the smaller one doing the compression 
and the larger performing the work. The displacement 
through the regenerator is performed by the pistons pass- 
ing the compressed charge from the smaller to the larger. 
Thermodynamically the Ericsson differs from the Stirling 
in that the regenerative processes joining the two isotherms 



i6o 



THE THERMODYNAMICS OF HEAT-ENGINES 



are under constant pressure instead of under constant vol- 
ume. The cycle is that portrayed in Figs. 29 and 30. The 
intake of fresh cold air into the smaller compressing cyl- 
inder is shown at aA, Fig. 30, and does not appear in 
Fig. 29. It is then compressed isothermally to B in con- 
tact with the refrigerating surface. It is discharged along 
Bb, Fig. 30, into the regenerator, through which it passes 
on its way to the larger working-cylinder. There it ap- 




FlG. 29 

pears as the volume bC, Fig. 30, having in the interim been 
heated along the constant-pressure isomorph BC of Fig. 29. 
At C the working-cylinder experiences a cut-off of supply 
exactly as in the steam-engine, after which isothermal ex- 
pansion takes place in contact with the fire-surface until 
atmospheric pressure is reached. 

Constructively and therm odynamically the Ericsson bears 
somewhat the same relation to the Stirling that the Joule 



THE HOT-AIR ENGINES 



161 



does to the Otto, except that the efficiencies of the two 
hot-air engines are the same, whereas those of the two 
gas-engines are not. Indeed, the diagram of the Joule 
engine, Fig. 23, will serve very well as an illustration of 
the general arrangement of the Ericsson engine if the 
chambers T x and T 2 be imagined as regenerators, each 
having its end toward C at the temperature T 2 and the end 




Fig. 30 



toward W at the temperature T v and if the cylinder C be 
imagined as jacketed with cold water and the cylinder W 
as surrounded with fire; for the working-cylinders of the 
Ericsson engines were actually hung over furnaces, much 
in similitude to huge camp-kettles. Another difference 
between the two hot-air engines was that in the Stirling 
the motion of the two pistons had to be properly timed, 
one with the other, to produce the proper effect, whereas 



1 62 THE THERMODYNAMICS OF HEAT-ENGINES 

in the Ericsson they were, within certain limitations, inde- 
pendent. 

Problem 31. What are the conditions of operation and 
the efficiency of an Ericsson engine working under the 
same temperature and pressure ranges as the Stirling 
engine of Problem 30? 

Here V A = 1.000. Since P B is to be 69.25, V B = 0.2097. 
Since T c = 2000 and the heating is under constant pressure, 

V c = x 0.2097 = 0.8073, and N c = 0.02417. The heat 

5 20 p j 

supplied along CD = C C T C \og e —■ • , from which Q 1 = 

"d 778 y j- 

16.18 B.t.u. and N CD = 0.00808. N D = 0.03225. ^f=4~, 
Y P VbT b 

whence Fc = 0.8073. — ^ = — £, whence V jD =S-^47- From 

Vq Pd 
the same argument as in Problem 30, the efficiency must 

T — T 

be given by * — a = 74.0%. The amount of heat con- 

^1 
verted into work, or Q w , = 0.74 x 16.18 = 11.97 B.t.u. 

The engines of the vessel Ericsson, the culmination of 
that engineer's efforts, well illustrate the reasons for the 
failure of the hot-air engine. Though designed to propel 
a vessel of 260 feet in length at the modest speed of seven 
miles per hour, the bulk and weight of the engine was 
enormous. It had four compressing-cylinders, each 11.4 
feet in diameter, and four working-cylinders, each 14 feet 
in diameter ; the stroke of each was 6 feet. The ves- 
sel sank before accurate tests of efficiency or capacity 
could be made, but as the maximum working temperature 
is mentioned as being about 443 , the theoretical efficiency 
could not be above 42.6%, and the actual efficiency must 
have been less than half of this, or not over 20%. As 
this is very much better than the best marine engines 
even of to-day can do, there is little wonder that the new 
"caloric " engine won tremendous applause as the superior 



THE HOT-AIR ENGINES 1 63 

of the steam-engine. But the difficulty of floating engines 
powerful enough to give the hull a practicable speed, even 
with the weight saved in coal-bunkers and boilers and the 
constant trouble and rapid deterioration due to attempting 
to lubricate the pistons under such (then) unusually high 
temperatures, were so obvious that the Ericsson, though 
promptly raised to the surface, was reengined as a steamer, 
and the attempt has never been repeated. 



CHAPTER VIII 

SUMMARY OF THEORETIC POSSIBILITIES OF 
VARIOUS TYPES 

The efficiency of a heat-engine is only one, and often a 
minor one, of the many factors which determine its value. 
The one always of great, and often of the first, importance 
is the capacity for developing power within a given volume, 
weight, or cost of machinery. Usually the qualities coming 
next in the order of importance are simplicity, reliability, 
and adaptability to varied service. But, confining the at- 
tention entirely to mere thermodynamic considerations, it is 
interesting to note the maximum possible efficiency of the 
several types when compared with what may be called the 
"builder's factor," viz.: the volume of cylinder (in cubic 
feet) times the maximum pressure upon the piston (in 
pounds per square inch), divided by the work developed 
(measured in heat-units), which is as accurate an indication 
of the cost per horse-power as can be based upon purely 
thermodynamic conditions. 

The higher the builder's ratio, the greater will be the cost 
of the engine per horse-power. 

This table shows clearly, if coupled with the statements 
of the obstacles arising in attempting, to utilize hot air, how 
those engines of the highest efficiency pay for it in bulk 
and cost, and, vice versa, how those of high capacity (or 
low cost) pay for it in poor efficiency. It will be noticed 
that all of the engines using permanent gases for working- 

164 



SUMMARY OF THEORETIC POSSIBILITIES 



I6 5 



Engine 



Builder's 
Ratio 



Steam-engine, non-condensing (Problem 13) . 

Steam-engine, non-condensing, modified by 
incomplete expansion ..... 

Steam-engine, condensing (Problem 15) . 

Steam-engine, condensing, modified by incom- 
plete expansion 

Steam-engine, non-expansive (Problem 19) 

Gas-engine, Lenoir (Problem 27) . 

Gas-engine, Otto (Problem 28) 

Gas-engine, Joule (Problem 29) 

Gas-engine, Joule, modified by incomplete ex- 
pansion 

Hot-air engine, Stirling (not including dis- 
placer-piston and regenerator) 

Hot-air engine, Ericsson (not including re- 
generator) 



16.0 

8.6 
54.0 

15.2 

5-5 

444 
3i-5 
47-9 

35-i 
6.4 

22.2 



substance exceed the non-condensing steam-engine in cost 
as well as in efficiency. 

It should be noted that all of the engines which expand 
to atmospheric pressure end the indicator-diagram with a 
sharp toe ; this calls for an exaggerated maximum volume 
without sufficient pressure,. toward the end, to develop com- 
mensurate power. This trouble is always obviated in prac- 
tice by ending the stroke with an appreciable pressure still 
in the cylinder, which allows the engine to make a better 
showing in regard to builder's ratio ; this stands quite aside 
from those considerations which affect all of the engines in 
passing from theory to practice. Thus, by ending the ex- 
pansion in Problem 13 at 15 pounds per square inch of net 
pressure the efficiency is reduced only from 14.9 to 13.35%, 
while the builder's ratio falls from 16.0 to 8.6. Similarly, 
ending the expansion in Problem 1 5 at 7 pounds per square 



1 66 THE THERMODYNAMICS OF HEAT-ENGINES 

inch net pressure reduces the efficiency only from 23.6 to 
21.2%, while the builder's ratio falls from 54.0 to 15.2%. 
Shortening the maximum volume of the Joule cycle to 
2.5 feet decreases the efficiency only from 60.7 to 57-7%, 
while the builder's ratio falls from 47.9 to quite near that 
of the Otto. 






CHAPTER IX 

THE REFRIGERATING MACHINES 1 

As in the case of the clockwise cycle, the reversed ther- 
modynamic or refrigerating cycle does not exist in practice 
in the rectangular or perfect form. Practical considera- 
tions make it more profitable to accept a lessened thermo- 
dynamic efficiency for the sake of a saving in complexity, 
cost, bulk, etc. 

The processes in actual use for the development of arti- 
ficial cold may be divided broadly into two classes : — 

I. Those depending upon the capacity of water for 
absorbing certain vapors at lower temperatures, which 
vapors may be driven off again by the addition of heat. 
They are called absorption-machines . They demand no 
supply of mechanical power, the heat-energy which keeps 
them in operation being supplied in the form of heat 
directly to the water, and they involve no more moving 
machinery than one or two circulating-pumps. They 
therefore exhibit no purely thermodynamic phenomena, 
and will be given no further consideration in these pages. 

II. Those operating on some thermodynamic cycle such 
as has just been discussed, and always involving a supply 
of power from without to keep them in operation. They 
are known as compression-machines . They may be sub- 
divided into two sub-classes : — 

(a) Those using a permanent gas, usually air, as a 
working-substance ; and 

1 For the general laws applying to the reversed cycle the student is referred 
to pages 55 to 62. 

167 



1 68 THE THERMODYNAMICS OF HEAT-ENGINES 

(J?) Those using a saturated vapor, usually that of anhy- 
drous ammonia, ether, carbonic acid, etc. 

Of these two sub-classes the air-machine is restricted 
almost wholly to use on shipboard. Of the vapor-machines, 
probably 95 % rely on anhydrous ammonia as their working- 
substance. 

The air-machines used purely for refrigeration always 
work on a reversed Joule cycle. Referring to Figs. 23, 
24, and 25, suppose that the shaft of the engine receives 
power from without which suffices to rotate it in a reversed 
direction. The direction of flow of the air, as indicated by 
the arrows, will also be reversed. In Fig. 25 let AE rep- 
resent a volume of atmospheric air taken in by the larger 
cylinder at a temperature of 520 absolute. Its thermo- 
dynamic condition is shown at E, Fig. 24. Let it be com- 
pressed adiabatically to D. Its temperature will be 1342 
absolute, or 88 1° F. Let the air now be passed through 
the chamber T v which has now become a " refrigerator " 
by being equipped with coils of pipe containing a circula- 
tion of cold water, until its temperature has been reduced 
to 59 F. again, or 520 absolute. Its volume at E being 
unity, that at D is 0.1039 and that at C is 0.0591. Let this 
compressed air under ordinary temperature now enter the 
smaller cylinder and reexpand there to atmospheric press- 
ure. Its volume will now be 0.584 cubic foot and its abso- 
lute temperature 204 , or 257 below zero Fahrenheit. It 
may now be exhausted into the room which is to be cooled 
at atmospheric temperature. 

Of course, in actual practice no such low temperature as 
the one just calculated is attained. The pressure-limits 
which were chosen as suitable for the heat-engine of Prob- 
lem 29 are far too great for the refrigerating-machines ; 
the cold produced is far greater than that desired for the 
preservation of meat, etc., and the machine itself would 



THE REFRIGERATING MACHINES 1 69 

freeze up in attempting to produce it. But the figures will 
illustrate how a Joule engine can be reversed for the pro- 
duction of cold. 

It is obvious from Fig. 25 that power must be supplied 
to keep the cycle in operation. The positive work devel- 
oped, the area cCBA, is much smaller than the negative 
work absorbed in compression, the area AEDc. As to 
efficiency, that depends upon the temperature of the re- 
frigerated room itself, which is represented in Fig. 24 by 
the ;trX-level. The heat taken from the room is that be- 
neath the curve BX. That beneath XE represents leakage 
of heat from atmospheric temperature into the working- 
substance. Again, the heat is discharged, along DC, into 
water which cannot have a temperature higher than the 
C-level ; therefore all of the area above that level repre- 
sents wasted work, — - similar to that of the man who sup- 
posedly lifts a barrel on his shoulder in order to raise it to 
a platform situated a foot above the ground. As BEDC 
represents the work done, the efficiency must be the area 
OBXz divided by the area BEDC, obviously a much poorer 
efficiency than the maximum possible, or that of a rectangle 
lying between the temperature-levels of C and X. 

The Ammonia Compression Machine. — The class of re- 
frigerating machines to which the ammonia compression 
machine belongs comprises the great majority of those 
used in industrial enterprise. All of the largest machines 
are of this type. Its action also typifies that of all other 
machines making use of a liquid refrigerant which is evap- 
orated into a saturated vapor and then returned to liquid 
form by compression and condensation. 

The operation of the machine is illustrated diagrammat- 
ically in Fig. 31. At the left is seen a "liquid-reservoir," 
which holds the stock of refrigerant (in this case liquid 
anhydrous ammonia) under a pressure sufficiently heavy to 



170 



THE THERMODYNAMICS OF HEAT-ENGINES 



eliminate all danger of vaporization under all ordinary- 
temperatures. The liquid exists, in fact, at atmospheric 
temperature, and may be carried in this condition in un- 
tagged pipes to any locality, however distant, without any 
loss of its refrigerative capacity. Leakage, of even the 
most minute character, must always be guarded against, 
on account of the penetrating and unpleasant character of 
the pungent odor of ammonia, and also on account of the 




Fig. 31 

expense involved in maintaining the stock of ammonia in 
the face of leakage. The liquid-reservoir is shown equipped 
with a gauge-glass whereby the condition of the stock of 
liquid may be seen at a glance. 

From the liquid-reservoir the ammonia passes, through a 
pipe known as the " liquid-line," to one or any number of 
expansion-valves, situated wherever refrigeration is desired. 
One of these expansion-valves is shown at AB. Upon 
escaping through this valve, which is usually a needle-valve 






THE REFRIGERATING MACHINES 171 

of quite fine construction, the liquid finds itself freed from 
the so-called " head-pressure " of something like 160 to 200 
pounds per square inch which kept it in its liquid condition. 
Beyond this valve it is under the comparatively light "back- 
pressure " of about 15 pounds by gage which prevails in 
the coil BC. The boiling-point of ammonia under this 
pressure is about zero Fahrenheit. In consequence, so 
long as the ammonia finds itself in contact with anything 
warmer than this temperature it will extract heat from it 
which is absorbed in the form of latent heat of vaporiza- 
tion. In exactly similar fashion a tea-kettle filled with 
water, finding itself in contact with a surface hotter than 
212 F., will abstract heat from it with which to vaporize 
the water ; nor can the metal of the tea-kettle be brought 
to a higher temperature than 21 2°, by the application of 
ever so much heat, so long as there still be water in it. 

In this way the ammonia cools the pipes BC so nearly 
as it can to zero Fahrenheit, and with them anything in 
contact with their outer surface. The coil BC may be 
situated in the room to be refrigerated, but more commonly 
it is placed in a tank of strong brine, as shown in Fig. 31, 
which is chilled to a temperature of io° to 20 F., and is 
then circulated through pipes in the rooms to be refriger- 
ated, or around cans containing pure water to be frozen 
into ice. The object in interposing the brine between the 
real refrigerating process and the object to be refrigerated 
is to obtain stability of temperature. The slightest touch 
of the valve AB will produce considerable variation in the 
refrigerative activity of the coil BC. With the great mass 
of brine present all such variations are absorbed without 
producing a correspondingly fluctuating effect upon the 
object to be refrigerated. 

So long as any liquid ammonia remains unevaporated in 
the coil BC it maintains its temperature at zero Fahrenheit. 



172 



THE THERMODYNAMICS OF HEAT-ENGINES 



Immediately upon complete evaporation, however, the 
higher temperature of the brine produces superheat, and 
this is increased by the heat of the atmosphere as the 
ammonia-gas passes from C to the compressor at D, a 
transit which may cover many feet of pipe. 

In the compressor DE the pressure of the ammonia-gas 
is raised from the back-pressure to the head-pressure again, 

E 



G/ Fl 




w/////////////////^^^^^ 



abg 



Fig. 32 



f.k c a 



and under this heavy pressure the hot, dry gas is passed 
through the condenser-coils FG, over which is running a 
circulation of cold water. The chill of the water first re- 
duces the temperature of the gas to the boiling-point under 
the head-pressure, and then, since this is always hotter 
than the water, condenses it back into its liquid form. 
From the condenser FG the liquid trickles down into the 






THE REFRIGERATING MACHINES 



173 



liquid-reservoir, or receiver, as it is sometimes called, where 
it awaits use over again. 

It is to be noted that, going in the direction of the 
flow of the ammonia, the back-pressure prevails from the 
expansion-valve AB to the compressor DE, while the head- 
pressure prevails from the compressor DE to the expansion- 
valve again. This divides the system between two distinct 
pressures. The first, or lower, is so regulated as to main- 
tain the proper temperature in the coil BC; this is done by 
adjustment of the expansion-valve and sometimes by vary- 




Fig. 33 

ing the speed of the compressor. The second, or higher, 
pressure depends upon the temperature and amount of 
cooling-water supplied ; it regulates itself more or less auto- 
matically, rising, under the influence of the discharge from 
the compressor, until the increased temperature makes the 
transfer of heat to the water sufficiently rapid to maintain 
thermal equilibrium. 

The thermodynamics of the process is shown in Figs. 32 
and 33. The point A represents the primary condition of 
the liquid as it approaches the expansion-valve AB. The 
passage through the valve, which consists of simple wire- 



174 THE THERMODYNAMICS OF HEAT-ENGINES 

drawing or throttling, is shown by the constant-heat curve 

AB. It is evident from this that a portion of the refrigera- 

tive power of the ammonia is consumed in reducing its 

own temperature to its boiling-point, before any can be 

available for refrigerating other bodies. At the point B, 

when the ammonia is first ready to chill the coil BC, the 

b' B 
proportion — — has already been vaporized and is incapable 
b C 

of further refrigerative activity. 

As the ammonia traverses the coil BC it is gradually 
evaporated under the constant back-pressure, as shown by 
the line BC. When vaporization is complete at C, which 
is usually accomplished before the ammonia leaves the 
coil, superheat begins. It ends as the gas reaches the 
compressor in the condition D. 

The next step consists of the adiabatic compression of 
the gas in the compressor, as shown by the line DE, the 
gas being discharged from the compressor in a condition 
of high superheat. 

As the gas flows from the compressor through the con- 
denser-coils its temperature is first reduced from E along 
the line EF, it is then condensed isothermally to liquid 
form along the line EG, and finally it is cooled as a liquid, 
during its stay in the receiver and on its way to the ex- 
pansion-valves, along the curve GA. The cycle is now 
complete. 

The efficiency, capacity, etc., may be calculated as in the 
other cycles. It is to be especially noted, however, that 
whereas in the heat-engine cycles the thing supplied is 
heat, or Q v and the thing wanted is work, or Q m and 

hence the efficiency is ^, now the thing supplied is work, 

or Q m and the thing wanted is refrigeration, or heat-ab- 
stracted, or Q 2 , and hence the efficiency of a refrigerating- 



THE REFRIGERATING MACHINES 



175 



machine is given by the expression —^ 



But since the 



former is not a portion of the latter, the two being quite 
independent quantities of energy, the ratio just expressed 
is not properly an efficiency at all, but a sort of coefficient 
of efficiency. It is none the less useful, and custom has 
warranted its continuance. 

To illustrate, let a set of conditions be assumed. 

Problem 32. What will be the amount of refrigerative 
effect and the efficiency of operation if one pound of liquid 
anhydrous ammonia at a temperature of 69 F. and a 
pressure of 210.7 pounds absolute be evaporated under a 
back-pressure of 29.74 pounds absolute, the temperature of 
the gas reaching the compressor being 39 F. ? 

Reference to Figs. 32 and 33 and to the ammonia-tables 1 
places at hand all of the necessary guides and data for cal- 
culating all of the heat-quantities desired. The values of 
abscissae and ordinates for the various points of the cycle 
are tabulated below. The manner of their calculation is as 
follows : — 



Point 


Pressure 
Lbs. p. sq. in. abs. 


Volume 
Cubic feet 


Temperature 
Fahr. F. Abs. 


Entropy 


A 


I04.gl 


O.IO9 


59 


520 


O.1325 


B 


29.74 


O.835 


O 


461 


O.140I 


C 


29.74 


9- 6 3 


O 


461 


I.2277 


D 


29.74 


10.44 


39 


500 


I.2689 


E 


2IO.7 


2.361 


34o 


801 


I.2689 


F 


2I0.7 


1.59 


100 


561 


I.0826 


G 


2IO.7 


0.109 


100 


561 


O.2161 


K 


29.74 


8.877 





461 


I.1317 



The cycle is most readily understood if assumed to start 

at A, where the pressure upon the liquid first becomes low 

1 See Peabody's " Steam-Tables," Table X, for instance. 



176 THE THERMODYNAMICS OF HEAT-ENGINES 

enough to permit vaporization. The path AB is a con- 
stant-heat curve of wire-drawing, whence the heat at B, or 
the area Ob'Bb, must equal that at A, or the area Ob'Aa. 
From the ammonia-tables the latter is found to be 64.6 B.t.u., 
reckoning from an arbitrary zero (for this problem) of zero 
Fahrenheit, at b' . Therefore, in Fig. 32, 

b'B = 64.6 -e-461 = 0.1401. 

The path BC represents the isothermal vaporization in the 
coils BC. The area Ob'Cc must represent the latent heat 
of vaporization, given as 566 B.t.u. ; whence b'C= 1.2277. 
The path CD represents the superheat on the way to the 
compressor and has the equation 

N-N c =S\og e ^- 

1 c 

The path DE shows the adiabatic compression in the com- 
pressor, and EFGA the cooling under constant pressure in 
the condensing-coils and in the liquid-lines. For these 
latter curves one must know the specific heats of ammonia 
under constant pressure : 1.095 for the liquid and 0.522 for 
the vapor, and the ratio of the two specific • heats for the 
vapor: 1.3 1 or 1.3 17. Since all of the conditions of the 
vapor studied in the present problem lie close to the satura- 
tion-point, and consequently the specific heats are not con- 
stant, the laws governing these curves will not be found in 
exact agreement, in their mathematical results, with the 
ammonia-tables ; but the variations will not be sufficient 
to affect the accuracy of the study from an engineering 
standpoint. 

The determination of the heat-quantities handled, Q v Q 2 , 
and Q w , requires little discussion. Q v the area a A GFEd, 
is readily found to be the sum of the heat of the liquid 
{aAGg— 45.4), of the latent heat gGFf( = 486.0), and of 
the heat of superheat fFEd{ = 157.0), or 688.4 B.t.u. Q m 



THE REFRIGERATING MACHINES 177 

the area ABCDEFGA, can only be found as the difference 
between Q 1 and Q 2 , the area aABCDd. In Q 2 occurs the 
area aABb\ reference to page 108 gives it as 4.3 B.t.u. 
The area beneath BC must be the latent heat of vaporiza- 
tion along b'C minus the heat of the liquid between b'A ; 
because the area Ob' Bb must equal the area Ob' Aa. The 
heat of superheat beneath CD is readily found from the 
specific heat and the range in temperature. 

From these data Q 2 is found to be equal to the sum of 
4-3, 501-4, and 20.3 B.t.u., or 526.0 B.t.u. This makes 
Q w = 162.4 B.t.u. But of this thermodynamic value of Q 
only the area bBCc is useful refrigeration ; therefore the 
efficiency-coefficient of the cycle is 

bBCc _ 501.4 _ 
ABCDEFG 162.4 ~ 3 ' 0y7 - 

It has been pointed out by Mr. William Lee Church that 
the efficiency of the cycle might be improved if the waste 
cold lost in the transfer of gas from refrigerating-coil to 
compressor, cCDd, might be used to chill the liquid to its 
boiling-point before it is released through the expansion- 
valve, by jacketing the pipe leading to the valve AB 
(Fig. 31) with the pipe CD. This would result in bringing 
the cycle down the path GAb' to the lower level instead 
of down the path GAB. Q 1 would now become 753 B.t.u., 
Q 2 586.4 B.t.u., and Q w 166.6 B.t.u. But the cold cCDd is 
not sufficient to absorb all of the heat OV Aa ; in order to 
carry frost visibly through this jacket-modification some 
liquid must be allowed to boil over into it, sufficiently to 
add to the cold an amount, kKCc, large enough to make the 
area kKCDd= Ob'Aa. From these considerations KC is 
found to be 0.0960, making b< K = 1. 13 17 and the area of 
useful refrigeration, Ob 1 Kk, = 521.7 B.t.u. The efficiency- 
coefficient is therefore now equal to 521.7-=- 166.6 = 3.130, 



178 THE THERMODYNAMICS OF HEAT-ENGINES 

a result some 1.4% better theoretically than that deduced 
above. In practice the gain is greater than this. 

The Series Refrigerating Process. — The ordinary indus- 
trial demand for artificial cold is for cold of a very moderate 
degree. The rooms of a cold-storage warehouse for the 
preservation of meats and other provisions which must not 
be frozen are kept as closely as possible at a temperature 
of 34°-36° F. Those where fish, butter, etc., are frozen 
need to be only slightly below 32^ F. Either of these 
duties are usually performed by a circulation of brine not 
colder than 20 F. and the same medium at the same tem- 
perature serves to freeze artificial ice. The brine is main- 
tained easily at this temperature by a refrigerant having 
no lower a temperature than o° F. 

But for scientific purposes the need of obtaining much 
lower temperatures than these has long been urgently felt, 
though the amount of heat to be handled is naturally small. 
For none of these purposes is the Joule air-cycle fit. The 
refrigerant is too diffuse. All of the successful methods 
which were formerly relied upon for very low temperatures 
made use of the ebullition under reduced pressure of some 
liquid whose boiling-point was unusually low even under 
atmospheric pressure. The locality to be refrigerated, 
usually a small piece of laboratory-apparatus, could be 
surrounded by the liquid and the precious cold conserved 
to the last degree. 

The liquids commonly used were carbonic acid or ether, 
operated upon the cycle of the ammonia-machine. But 
even under vacua these liquids would not absorb heat at a 
sufficiently low temperature. Recourse was then had to 
a thermodynamic series. Liquid carbonic acid, boiling 
under reduced pressure, would produce a degree of cold 
which, aided by high pressure, would condense some more 
refractory vapor, its heat being absorbed by the boiling 



THE REFRIGERATING MACHINES 1 79 

carbonic acid. This refractory liquid, when once the press- 
ure upon its surface were relieved, would boil and absorb 
heat at a much lower temperature than would the carbonic 
acid. This process is the thermodynamic obverse of the 
Series heat-engine. 

By this means many of the so-called permanent gases 
were liquefied in scientific laboratories and much of inesti- 
mable value was accomplished. But the limitations of the 
process were found, in complication, expense, and rapidly 
diminishing efficiency, and rapidly increasing labor with 
which even the smallest quantity of heat could be ab- 
stracted at the lowest temperatures, at a point which still 
fell far short of attaining degrees of cold for which scientific 
investigators felt urgent need. The entire situation has 
now been solved by the Linde liquid-air process, which 
can of course be applied to the liquefaction of other 
"permanent" gases almost as readily as it can to that of 
atmospheric air. 

Lord Kelvin's Warming-machine. — It has been pointed 
out by Lord Kelvin that whereas, in the ordinary refriger- 
ating-machine, the absorption of the heat Q % is the thing 
desired, the development of the discharged heat Q x being 
an undesired incident in the process, this aspect of the 
cycle might be reversed by applying Q x to the heating of 
buildings. Indeed, the machine might be wholly devoted 
to this purpose, in which case Q 1 becomes the end in view 
and Q 2 becomes the incidental factor. Thus, instead of 
applying the heat developed from burning coal directly to 
the rooms to be heated, this heat might be used in a heat- 
engine to develop power, the exhaust from the heat-engine 
being utilized for heating the building ; the power so 
developed might then be utilized in driving a refrigerating- 
machine which absorbed low-temperature heat from out-of- 
doors and discharged its high-temperature heat into the 



l80 THE THERMODYNAMICS OF HEAT-ENGINES 

building. The proposition might be paralleled, for eluci- 
dation, to that of a man wishing to fill a land-locked basin 
lying some few feet above the sea-level with water and 
having, as a means with which to work, a small stream 
falling from a mill-pond situated ten times as high as the 
basin above the sea. He might, in the first place, simply 
let the stream flow into the basin, which it would eventually 
fill. This would correspond to the ordinary method of 
heating buildings. But if his small water-power were 
extremely valuable, or if time were valuable also, he might 
hasten matters by letting the small stream, in its fall into 
the basin, drive a water-wheel. By attaching to this water- 
wheel a pump piped to lift water from the sea and to dis- 
charge it into the basin, it is plain that the latter would be 
filled much more quickly and at a much less cost of mill- 
pond water. If the basin were 10 feet and the small mill- 
pond ioo feet above the sea-level, respectively, the basin 
would be filled in the second case in one-tenth of the time, 
or with one-tenth as much water drawn from the mill-pond, 
as compared with the first case. To illustrate the proposi- 
tion thermodynamically the following problem will be 
useful : — 

Problem 33. What will be the comparative amount of 
heating accomplished (1) by applying a given amount 
of heat directly to a building, and (2) by making use of a 
Kelvin warming-machine made up of the steam-engine of 
Problem 14 driving the ammonia compression machine of 
Problem 32 ? (In this, the temperature out-of-doors is sup- 
posed to be that of the BC-\evel of Fig. 32.) 

According to Problem 14, for every 100 B.t.u. supplied 
to the engine 23.6 B.t.u. are transformed into work. The 
remaining 76.4 B.t.u. may be imagined as exhausted into 
the building at a temperature of I3I°F. According to 
Problem 32 (2d Case) the supply of 23.6 B.t.u. of work 



THE REFRIGERATING MACHINES l8l 

will absorb from the lower level of zero Fahrenheit and 
discharge at the higher level of ioo° F. : 3.13 x 23.6 
= 73.9 B.t.u. of low-temperature out-door heat. This, 
added to the 23.6 B.t.u. of work which is converted into 
heat within the refrigerating-machine and to the 76.4 B.t.u. 
of exhaust-heat from the engine, makes a total of 173.9 
B.t.u., which reach the building from the original supply 
of 100 B.t.u. The efficiency of the fuel has been increased 
by 73.99b. 

A little consideration will show that this effect does not 
stand as an infraction of the law of the conservation of 
energy, as may at first appear, but as a combination of the 
operation of that law within the heat-engine with the opera- 
tion of the second law of thermodynamics in the refriger- 
ating-machine. No heat has been created. Heat has been 
merely lifted up-temperature from out-of-doors into the 
building. 

This plan may be best imagined as put into practice by 
heating the building with hot-water pipes filled with the 
circulation coming from both the steam and the ammonia 
condensers. It is, of course, impracticable at present by 
reason of the great cost of apparatus and of attendance. 
But it is by no means improbable that at some time during 
the present century, when coal shall have become much 
dearer and machinery much cheaper and more simple than 
now, the plan may be relied upon for the heating in winter 
(and the cooling in summer) of our larger buildings. 



PART II 

APPLICATION OF THEORY TO 
PRACTICE 



CHAPTER I 

THE SIMPLE STEAM-ENGINE 

- With a full understanding of the theoretical laws which 
limit and control the development of power from heat in 
the various types of engines in use, it is next in order to 
discuss in what points the operation of these theoretically 
perfect engines are modified by the limitations of actual 
construction and by those thermal phenomena, such as the 
conductivity of the cylinder-walls, etc., which have been 
carefully excluded from the foregoing analyses. 

The theoretical cycles are modified by these factors in 
two distinct ways either of which may be of prime impor- 
tance in any given case : — 

(i) In Efficiency, or in the proportion of the heat theo- 
retically available for work which is actually developed ; 
and 

(2) I11 Capacity, or in the amount of power available 
from a given size or weight of engine. 

The first characteristic is customarily expressed, in re- 
gard to any given engine, by a ratio called its cylinder- 
efficiency. Following the notation of the first portion of 
the book, this cylinder-efficiency may be defined as follows : 
Let the heat supplied to the engine be represented by Q 1} 
that abstracted by the refrigerator by Q 2 , that theoretically 
available for conversion into work by Q w , and that actually 
converted into work by q w ; then the expression for the cylin- 
der-efficiency would be — 

27 Qw 



Q w <2i - 02 
185 



1 86 THE THERMODYNAMICS OF HEAT-ENGINES 

This expression is a valuable one for the indication of the 
proper place of the engine upon the efficiency-records. It 
covers the skill with which the designer and the builder 
together have been able to develop within the cylinder a 
proper portion of the work which the thermodynamic limi- 
tations permitted them to expect. It sweeps away all the 
complexity and ambiguity involved in attempts at deduction 
from the varied conditions of actual operation to a common 
basis of comparison for engines of any sort. Not only do 
the most diverse types of steam-engine fall into line for such 
equity of comparison, but gas-engines or hot-air engines 
lend themselves to the same process with equal docility and 
permit themselves to be placed thereby upon the same 
platform with the steam-engines without injustice to either. 

For instance, it happens that a common value of cylin- 
der-efficiency for simple single-cylinder steam-engines and 
for Otto gas-engines is 65%. That is, of the work theo- 
retically available in each case the designer is able to 
develop about two-thirds. This proportion sometimes rises 
above 70% ; it more frequently falls to 60%, or even below. 
It is also an invariable rule that this proportion varies in- 
versely with the engine's complexity. That is, it is highest 
in simple engines, lower in double compound, and lowest in 
triple compound engines. 

The only other factor needed to express the value of the 
engine as a means of development of power is the efficiency 
with which the work developed within the cylinder is trans- 
mitted to some useful destination, usually the belt. The 
loss in transmission is, of course, due solely to engine- 
friction. The ratio of work arriving at the belt to that 
developed upon the piston is referred to as the mechanical 
efficiency of the engine. If we represent the work thus 
transmitted to the belt by g b , the several efficiencies might 
be expressed as follows : — 



THE SIMPLE STEAM-ENGINE 1 87 

(1) Cycle-efficiency = =~ ', 

(2) Cylinder-efficiency = -jy ', 

(3) Thermodynamic efficiency — ~ or -^ ; 

(4) Mechanical efficiency = — • 

Only the last three are ordinarily needed or used. As to 
the thermodynamic efficiency, there is a lack of unanimity 
as to whether the belt-work or the indicated work should 
form the basis for comparison; but the only grounds for 
adhering to the latter are a custom grown up from an 
unfortunate mistake in the past. All of the arguments are 
in favor of referring to belt-power only. If this is done, 
the cylinder-efficiency and the thermodynamic efficiency 
cover all needs for identifying the engine's rank as a trans- 
former of heat into useful energy. 

As to capacity, there is no accepted form for the expres- 
sion of the engine's rank relatively to other engines. 
A common one is to state the number of horse-power 
developed per ton of machinery. This definition usually 
includes the boilers and all accessories, and is very useful 
in marine engineering. But for the study of the science 
of engine-design it is too comprehensive. The number of 
pounds weight of actual engine per horse-power is also 
widely used. The only objection open against it as a char- 
acteristic of capacity is that it gives no idea as to either 
space or cost involved. In the writer's study of the gas- 
engine problem he has needed to bring engines of the 
most diverse types into some comparison which would 
include all of these items in a way at least approximately 
just, and for that purpose has made use of what was 
defined as the builder s ratio in Chapter VIII, Part I. It 



1 88 THE THERMODYNAMICS OF HEAT-ENGINES 

was found to be of great service for this purpose ; but the 
practice is not yet a matter of general acceptance. 

Before entering upon a general discussion of the modifi- 
cation of theory in practice, however, it is necessary to 
acquire some familiarity with the machinery, so to speak, 
of engine-design, of the graphical methods whereby the 
actual engine-cylinder is brought into close touch with the 
purely theoretical diagrams which have been discussed 
hitherto. This can best be done by entering upon a given 
problem of engine-design. 

Problem 34. Wanted : An engine to develop 100 horse- 
power under a boiler-pressure of 100 pounds by gage, when 
exhausting to the atmosphere and running at a speed of 
100 revolutions per minute. 

The choice of the items just specified might call for the 
exercise of judgment on the part of the designer or they 
might be determined for him by existing conditions. In 
either case, immediately the problem is entered upon, his 
judgment is called upon to decide upon what type of engine 
the design is to be founded. 

The wisdom of this method of procedure needs more 
emphasis than mere passing mention. The chief charac- 
teristics in operation which determine the worth or the un- 
worth of an engine are determined, not by the detail of 
design and construction which the engine embodies, but to 
the general type or class to which it belongs, and to the 
correspondence between the type chosen and the service to 
be performed. It is too widespread an idea that the suc- 
cessful design of a steam-plant consists in the discovery of 
the "best" engine, the "best" boiler, the "best" pump, 
etc. This is far from right. There is no literally best 
engine, etc. There are many better ones and many worse. 
There are some so bad that they would be worthless any- 



THE SIMPLE STEAM-ENGINE 1 89 

where ; but there are none so good that they will not be 
worthless if located unwisely and set to perform work for 
which they were not designed. It is for the identification 
of the points in which engine and service should harmonize, 
and their recognition in practice, for which consideration 
must be turned to the question of the broad types of engine. 
If the wrong type be selected, for purchase or for design, 
no amount of skill in detail can produce good, or at any 
rate, the best, results. 

The distinguishing characteristics between the several 
types of engine, excluding questions purely of form, such 
as vertical vs. horizontal, etc., depend almost entirely upon 
the class of valves and valve-gear with which each is pro- 
vided. The study of the kinematics of valve-gears is no 
part of the work in hand ; but the following analysis of 
their mechanical and thermodynamic effects is essential. 

All cylinder-and-piston steam-engines come under classi- 
fication into three distinct types, each of which has its plain 
characteristics : — 

I. The single-valve fixed-eccentric type ; 
II. The four-valve fixed-eccentric type; 
III. The single-valve variable-eccentric type. 

Type I. This finds its typical illustration in the plain 
slide-valve throttle-governed engine, now almost obsolete 
for general power purposes. Its typical indicator-cards are 
shown in Fig. 34, the full line representing normal load, the 
dotted line overload, and the broken line light load. The 
resultant characteristics of the engine, when compared with 
those of the other types, may be stated relatively as follows : 

1 . Steam-distributioti : All events fixed. 

2. Efficiency : The poorest. 

3. Capacity by unit of space or weight : The greatest. 

4. Speed: Not limited by valve-gear ; usually medium. 



190 



THE THERMODYNAMICS OF HEAT-ENGINES 



5. Standard Ratio of expansion : 1.2 to 1.4. 

6. Clearance: Maximum. 1 

7. Regulation: Poor. 

8. Complexity : The least. 

9. Cost : The least. 



Boiler Pressure 




Fig. 34 

It may sometimes be found modified in the following 
particulars, without effect upon its characteristics : — 

(1) Piston-valve instead of slide-valve; 

(2) Hand-control of speed instead of governor-control. 
It is now used chiefly for portable work, such as hoisting, 
pile-driving, etc., or in very small powers. Even for small 
powers, however, if the service be a continuous one, engines 
of Type III may better be substituted. 



Boiler Pressure 




Back Pressure 

Fig. 35 

Type II This finds its typical illustration in the standard 
Corliss engine, with centrifugal governor and drop cut-off 
gear. Its typical indicator-cards are shown in Fig. 35. 

1 For definitions of " clearance " and " cylindrus " see page 205. . 



THE SIMPLE STEAM-ENGINE 19I 

The resultant characteristics, compared with those of the 
other types, are : — 

Steam-distribution : Cut-off variable by governor ; ex- 
haust, compression, and admission fixed. 

Efficiency: Above about 100 horse-power, maximum; 
below about 100 horse-power, medium. 

Capacity per unit of space or weight : Minimum. 

Speed: Limited by valve-gear to about 100 r.p.m. ; usu- 
ally less. 

Standard Ratio of expansion : 5 to 6. 

Clearance: Minimum. 
. Regulation : Poor to medium. 

Complexity: Maximum. 

Cost: Maximum. 

This type is found under very wide modifications with- 
out effect upon its characteristics. Any engine in which 
the steam-distribution is as stated belongs to this class, 
although the valves may be slide-valves, as in the Greene, 
or poppet-valves, as in the Putnam, or gridiron-valves, as 
in the Wheelock, or Myers cut-off double-slides, as in 
the Buckeye. It may have two fixed eccentrics instead 
of one. 

If the steam-valves, although Corliss in form, are actuated 
by positive connection to a variable eccentric in a shaft- 
governor, as is now done in some designs of Corliss engines 
in order to permit high rotative speed, the engine belongs 
not to Type II but to 

Type III, which finds its typical illustration in the high- 
speed, so-called " automatic," shaft-governor engine, usually 
fitted with a single piston-valve. Its typical diagrams are 
those of Fig. 36. The resultant characteristics, compared 
with those of the other types, are : — 

Steam-distribution: All events varied with the cut-off 
by the governor. 



192 THE THERMODYNAMICS OF HEAT-ENGINES 

Efficiency: Above about ioo horse-power, medium; 
below about ioo horse-power, maximum. 

Capacity: Medium. 

Speed: Not limited by valve-gear ; usually high. 

Standard Ratio of expansion : 4. 

Clearance: Medium. 

Regulation : The best. 

Complexity : Medium. 

Cost: Medium. 

Amongst the possible modifications of this type the one 
of the first importance is that involved in the substitution 
of the Stephenson-link or similar reversing-gear in place of 

i\ Boiler Pressure 



Fig. 36 

the variable eccentric, for they are virtual equivalents. 
This brings all locomotive and all marine engines, except 
the old-fashioned walking-beam paddle-engine, into this 
class. Any engine having a slide-valve, balanced or un- 
balanced, single or double ported, or divided into two or 
four parts, if driven from a variable eccentric, comes within 
the same classification. 

To return to Problem 34, it may be assumed that the 
engine is to belong to the Type II class, with a normal 
cut-off at one-fifth stroke. For while the text-book must 
emphasize the importance of selection according to type- 
characteristics, it can give no rules for making a wise 



THE SIMPLE STEAM-ENGINE 1 93 

selection. That must depend upon judgment cultivated 
by experience. 

The first duty is to construct a preliminary hypothetical 
indicator-card upon which to base estimates. This will, in 
the first instance, take the simple form ABCDE, Fig. 37. 
The pressure at AB is 100 pounds by gage, or 114.5 abso- 
lute ; that at DE is atmospheric plus a proper allowance 
for back-pressure, say, 16 pounds absolute. The volumes 
are unknown ; but ED = 5 AB. 

In the expansion of saturated steam in an engine-cylin- 
der the supposed adiabatic is always modified by heat- 
interchanges with the cylinder-walls. These interchanges 
are quite complex, and vary widely in different cases. They 
will be discussed more elaborately at a later point in the 
text. Taken all together, however, under average condi- 
tions, they usually have the effect of bringing the expansion- 
curve away from a true adiabatic and into approximation 

P V 
with the equilateral hyperbola having the equation — 1 = — 2 - 

2 ^1 
So, for the purposes in hand, it is always considered 

sufficiently exact (and it is certainly very convenient) to 

adopt this curve as being the same as BC of Fig. 37. 

N.B. This equation is identical with that for the isotheritial curve 
with air or other permanent gas as the working-fluid. But this fact 
must not lead to confusion with the conditions which prevail when 
steam is the working-fluid. The above equation does not represent an 
isothermal with steam, as is often erroneously stated. The isothermals 
in the steam-engine cycle are the lines AB and DE, and their equation 
is P= a constant. 

. For the drawing of the equilateral hyperbola BC, Fig. 37, 
there is a very convenient graphical method. Although 
the data of the problem under discussion do not give any 
direct information as to volumes, and it is therefore impos- 
sible to draw the curve to scale, at this point in the devel- 
opment of the problem, yet it will be convenient to have 



194 



THE THERMODYNAMICS OF HEAT-ENGINES 



before the eye an illustrative curve of indefinite scale upon 
which to base argument. Taking any initial volume at the 
stated boiler-pressure, as at the point B, the remainder of 
the curve may be determined in this way : — 




FIG. 37 



Let it be desired to know the pressure which will prevail 
at some other volume than the initial, such as at vv. 

( i ) Draw the constant-pressure line Bb until it intersects 
vv at b; 

(2) From b draw a diagonal bO to the absolute zero- 
point of pressure and volume ; 

(3) From B draw a constant-volume line Bm until it 
intersects bO at m ; 

(4) From m draw a constant-pressure line mn until it 
intersects vv at n. n is the desired point through which 
the curve is to be drawn. 

For finding an unknown point at any desired pressure 
other than the original, interchange all the italicized words 
"pressure" and "volume" in the above. Volumes or 
pressures may be greater or less than the original without 
altering the rule to be followed. 

The area of the cycle of Fig. 37 may be conveniently 
divided into two portions : ABCF and FCDE. Let the 



THE SIMPLE STEAM-ENGINE 1 95 

volume at C '— V c . Then area ABCF = P B V B \og e r = 
(114.5 x 144) x — - x logg 5. The area FCDE — FC x CD 

— ^[(5 x 1 14.5) — 16] x 144 = 6.9 x V c x 144. Since the 
engine is to develop 100 horse-power at 100 revolutions, or 
200 strokes, per minute the work necessarily developed per 
stroke, in foot-pounds, must be 

ttt 100 x 33000 

200 ~~~ l ®>$°? foot-pounds. 

Hence (114.5 x 144 x J V c x 1.609) + ( 6 -9 x x 44 x Vc) 

= 6sooV c = 16,500, 
or V c = 2.620 cubic feet. 

This result is of no immediate value, because no allow- 
ance has been made for the effect of clearance and com- 
pression. In professional engineering offices this allowance 
is usually made by judgment, as a percentage in addition 
to the calculated volume ; and since the result desired is 
nothing more accurate than the general dimensions of a 
cylinder of a capacity which is entirely nominal, this is 
usually amply accurate. But where the necessary experi- 
ence as a basis for estimate is lacking, clearance must be 
allowed for in the following way. In compound engines, 
too, where it is desired that a certain effective cylinder- 
volume be attained in order to establish a certain effective 
cylinder-ratio, it is important that the effect of clearance 
be carefully considered. 

Let the probable clearance to be expected in the cylinder 
when completed be estimated at 2 per cent. Suppose that 
it be the designer's judgment that this clearance should be 
filled with steam of two-thirds boiler-pressure at the com- 
pletion of compression. The situation is shown in Fig. 38. 

For the moment call the volume of the cylindrus, or fC, 



196 



THE THERMODYNAMICS OF HEAT-ENGINES 



equal to unity. Then Ff= 0.02 and FC = 1.02. AB must 
equal 0.204, in order to maintain a ratio of expansion of 5. 
Since the V c of Fig. 38 is equal to 1.02 times fC, it may be 
stated, from Fig. 37, that the area aBCDE of Fig. 38 is 

P 

hk B 




Fig. 38 

equal to 1.02 x 6300/6" = 6426/C. The area hHGE must 
be equal to P H V H \og e (P H -=- P G ) 

= Q 1 14.5 x 144) 0.02 fC log e (J x 1 ~ L ^j = 343 fC. 

The area aAHJi = 1 x 114.5 x 144 x 0.02 fC = no fC. 
Therefore the area ABCDGHA — (6426 — 343 — iio)/C 
= 5973 fC. Therefore, by the premises of the problem, 
5973/^= 16,500 ft.-lbs. ; whence fC= 2.76 cubic feet. 

Piston-speed. — This volume can of course be arranged 
in any form that the designer prefers : a long narrow cylin- 
der or a short and wide one. This would ordinarily be 
determined by the piston-speed, which is arbitrarily defined 
as the total distance travelled by the piston in feet per 



THE SIMPLE STEAM-ENGINE 1 97 

minute. It is usually considered desirable to keep this 
speed between 400 and 800 feet per minute, the larger 
figure for large engines and the smaller one for the little 
ones. In the present problem a piston-speed of 450 feet 
per minute would give a stroke of 27 inches and a diame- 
ter of 15 inches, in order to displace 2.76 cubic feet per 
stroke. In practice the stroke would probably be made 
28 inches, rather than 27. 

Mean Effective Pressure. — The mean effective pressure 
in an engine-cylinder is defined as that pressure which, if 
evenly maintained throughout the entire stroke, would accom- 
plish the same work as the actual varying pressure. 

The writer's experience has gone to show that for the 
ordinary purposes of a preliminary estimate, lumping all 
the questions as to probable back-pressure, cushion, and 
clearance to be incurred under a single coefficient based 
upon the type of engine, the following formulas are suf- 
ficiently accurate and are much quicker to handle than the 
plan just described. If P be the given boiler-pressure (by 
gage) then the mean effective pressure, p, can be had as 
follows : J — 



For Type I : p = 0.7 P — 6 
For Type II : p = 0.4 P — 2 
For Type III : p = 0.6 P - 8 



or / = §(/> + 15)- 20. 
or p = ±(P + 15)- 17- 
or / = §(/>+ 15) -23. 



The second set of formulas are somewhat more accurate 
and also more cumbrous than the first set. Since accuracy 
in such a case is a purely comparative term it is proper to 
add that the first set becomes unsatisfactory only when 
extreme conditions of low boiler-pressure prevail or when 
a vacuum is used. 

1 These formulas are also useful for estimating the proper capacity of an 
engine the records of which are wanting. 



198 THE THERMODYNAMICS OF HEAT-ENGINES 

Application of the first formula for Type II to the pres- 
ent problem gives ^=3.015, or a cylinder measuring 
15" x 30", which is on the safe side of the more exact 
figure determined above. The second formula gives a 
cylinder 15" x 28", which agrees exactly with the dimen- 
sions calculated above. At pressures lower than 100 by 
gage, which is unusually high for a simple engine, the 
agreement between the results from the empirical and the 
exact formulae would be closer. 

With this preliminary outline of the relations between 
pressure-volume curves and the actual cylinder, attention 
may be intelligently returned to the thermodynamics of 
engine-action. The discussion of the steam-engine cycle, 
which was left in its purely theoretic aspect, may now be 
carried on into its actual aspect in practice. 

The Effect of the Cylinder-walls upon Efficiency. — The 

Rankine cycle, the cycle of the perfect steam-engine, is 
impossible of reproduction in practice. The suppositions 
upon which its theory is founded are : — 

(1) No heat-interchange between the steam and the sur- 

rounding walls ; 

(2) Complete expansion to the lowest available pressure 

and temperature; 

(3) Instantaneous action of valves ; 

(4) No leakage past valves or piston ; and 

(5) No clearance. 

None of these conditions is realizable in practice except 
the second, and that is seldom desirable for practical 
reasons. (See pages 234-235.) 

Condensation and Reevaporation. — Of the five sources 
of variation of the actual cycle from the theoretic, the first 
is by far the most important in its effects upon efficiency. 
It may be remembered that the work which placed the 



THE SIMPLE STEAM-ENGINE 1 99 

name of Watt at the head of the list of steam-engineers 
was the study of this phenomenon. Impressed by the 
large amount of steam consumed by a small model engine 
submitted to him for repair, he attained its explanation by 
the following series of determinations : — 

1. The proof beyond doubt that a greater volume of steam 
entered the cylinder at each stroke than the cylinder conld 
possibly coyitain. Since the only possible method of meas- 
uring the quantity of steam entering the cylinder at each 
stroke was in terms of its weight, which could be measured 
before the water entered the boiler, Watt's task involved 
the determination of the specific volume of steam. His 
first approximation gave 1 : 1800 as the ratio between vol- 
umes of a given weight of water when liquid and when 
vapor. This led to the rule-of-thumb, current for long 
afterward among steam-engineers : "A_ cubic inch of water 
makes a cubic foot of steam," which implies a ratio of 
1 : 1728. This, of course, was all based upon steam of 
atmospheric pressure. We now know that the true ratio 
is about 1 : 1660. Knowing the weight of water evaporated 
in the boiler for each stroke of the engine, Watt knew 
from this ratio its volume. He was surprised to find it 
some four to six times that of the cylinder itself. In other 
words, the cylinder-condensation amounted to from 75 to 
83$. 

2. Watt proved that this condensation was not due to 
radiation of heat from the external surface of the cylinder. 
More accurately, these proofs were reversed in order; 
that is, suspecting condensation from general considera- 
tions, and radiation being the readiest explanation of con- 
densation, Watt first proved that the prevention of radiation 
was no remedy for the trouble. As this result would lead 
the ordinary mind to the conclusion that there could not 
have been any condensation, it is significant of Watt's 



200 THE THERMODYNAMICS OF HEAT-ENGINES 

genius that at this point in his investigations he persisted 
in the difficult task of proving conclusively that condensa- 
tion did take place, and to an enormous amount. If the 
cylinder-walls did not take up the heat released by this 
condensation and radiate it, what could pick it up and 
carry it away ? The only other thing in the cylinder 
besides the metal was water, and this hint led him to his 
next step. 

3. The determination of the latent and specific heats of 
steam and water, which showed that even a slight portion 
of the water injected into the cylinder at each stroke for 
the purpose of condensation was quite able to account for 
the trouble. To prevent it, therefore, could obviously be 
prescribed Watt's famous rule, easily deduced from what 
had preceded : — 

4. "Keep the cylinder dry, and as hot as the enteri7ig 
steam." To accomplish the first, one must use a separate 
condenser. For the second was needed the steam-jacket. 
Both, but especially the first, have survived until the present 
day as the first essentials to steam-engine economy. 

It was but a slight step in advance of this to recognize 
that the cylinder- walls as well as the water upon them were 
active in losing heat from the entering steam to the exhaust. 
It had been proven beyond doubt that it was not loss of 
heat through the walls to the air, but the alternating pas- 
sage of heat from steam to metal and water, and then back 
again from metal to steam, which lay back of the greatest 
loss of efficiency in the steam-engine. This is still the fact 
to-day, in spite of the century of effort at minimizing the 
thermal action under discussion. Its determination at that 
early date classed Watt as a scientist as well as a skilled 
mechanic, and in both fields of activity a genius. 

When boiler-steam enters an engine-cylinder, at or just 
before dead-centre is reached, it finds itself within a disk- 



THE SIMPLE STEAM-ENGINE 201 

shaped space enclosed by the cylinder-head, the piston and 
a narrow strip of cylindrical barrel-surface. This space is 
also one with that of the ports and passages for admission 
and exhaust of steam. In both its portions this steam-filled 
cavity presents a very great proportion of surface of sur- 
rounding wall relatively to its volume. This surface, too, 
for reasons which have already been outlined and which 
are to be elaborated later in this discussion, is always at a 
temperature much below that of the entering steam. The 
transfer of heat from a moist vapor to iron is very rapid. 
Therefore considerable steam condenses upon the surface 
of the metal, at the same time heating the latter to approxi- 
mately its own temperature to a certain depth, usually a 
very small fraction of the thickness of the walls. 

The steam thus condensed into water deposits upon the 
surface of the metal in bead-like or sweat-like drops. Since 
the transfer of heat between water and iron, or between 
steam and water, is still more rapid than that between 
steam and iron or between iron and iron, the presence of 
this water upon the surface of the metal very much exag- 
gerates the phenomenon just described. In fact, in many 
of these cylinder-phenomena the water adhering to the 
cylinder-walls is much more active as a thermal agent than 
is the iron, which is to be explained by the fact that the 
specific heat of water is much higher than that of iron. 

It is to be especially noted that the greater portion of 
all of the surface which is at any time brought to steam- 
temperature is exposed to the steam before the piston moves 
out upon its stroke, and this for a comparatively long 
period of time. This means that the bulk of the condensa- 
tion occurring in the cylinder does so while the engine 
is on dead-centre. It is this fact which primarily ac- 
counts for the need of lead in a steam-engine valve-gear. 
Something like 20 to 40% of all the steam used is wanted 



202 THE THERMODYNAMICS OF HEAT-ENGINES 

before the stroke begins. The one other basis for lead — 
the need for having the valve appreciably open when the 
piston starts on its stroke with accelerating velocity — is 
entirely a secondary one ; for in every case the valve-open- 
ing would be accelerated more rapidly than, and would 
attain a maximum before, the velocity of the piston does 
the same, even if there were no lead. 

As the piston moves out upon its stroke more and more 
cylinder-surface is exposed to boiler-temperature. Con- 
densation continues, but at a much reduced rate. When 
cut-off takes place the piston is usually moving quite rapidly 
and the exposed surface is increasing correspondingly. 

As soon as cut-off is effected, however, expansion ensues 
and the temperature of the steam-and-water mixture within 
the cylinder necessarily drops with the pressure. The bulk 
of the metal in the cylinder-walls, too, is not at the tempera- 
ture of the atmosphere, but at some mean between that and 
boiler-temperature, determined by the equilibrium which 
is established between the various gains and losses of heat 
without and within. The temperature-difference between 
the steam .within the cylinder and the fresh surface un- 
covered by the motion of the piston rapidly decreases, 
therefore, after cut-off takes place, and with it decreases 
the transfer of heat between the two. The rate of con- 
densation, which had steadily decreased as the piston ad- 
vanced, soon becomes zero. At this point in the stroke 
the proportion of water to total weight of steam and water 
within the cylinder may be anywhere from 10 to 60%, 
according to the size of cylinder and conditions of opera- 
tion. At cut-off it is seldom more than 5 or 10% less 
than this. 

The temperature of the expanding steam has no sooner 
fallen to the mean temperature of the iron than further ex- 
pansion, or the opening of the exhaust-valve, carries it be- 



THE SIMPLE STEAM-ENGINE 203 

low. The temperature-difference between the two becomes 
reversed, the direction of heat-transfer is reversed, and a 
reevaporation of the moisture deposited upon the cylinder- 
walls takes the place of the previous condensation. This 
continues, being very much accelerated by the opening of 
the exhaust-valve and the consequent reduction of pressure 
upon the hot water, until the latter is completely evaporated 
and the walls are dry. After that the rate of heat-transfer 
between metal and steam is reduced to a very small fraction 
— ■■ something like one two-hundredth, according to some 
authorities — of what it was while the walls were wet. 

The reevaporation of the water from the cylinder-walls 
of course demands heat for its performance, and the ab- 
straction of this heat from the metal lowers the temperature 
of its inner surface practically to that of the exhaust-steam. 
It is this abstraction of heat by reevaporation, and not the 
loss of heat to the outer air by external radiation or conduc- 
tion, which makes the demand for initial condensation 
upon the next admission of steam. Comparing these two 
sources of dissipation of heat : radiation and reevaporation, 
the former amounts to not over i to 10% of the latter. It 
is usually about 3% in single-cylinder non-jacketed engines. 

The Entropy-temperature Analysis 

This double process, condensation and reevaporation, is 
the first and greatest cause of the departure of actual steam- 
engine efficiency from the maximum possible determined 
by the external conditions (such as boiler-pressure, vacuum, 
etc.) under which the engine is forced to operate. The 
reduction of this loss has been the chief anxiety of the 
steam-engine designer since the days of Watt, and the 
amount of its reduction has been one of the best measures 
of progress accomplished. The effect of purely external 



204 THE THERMODYNAMICS OF HEAT-ENGINES 

conditions upon engine-efficiency is so well understood, the 
comparative cost and gain to be associated with a given 
rise of steam-pressure or decrease of back-pressure are so 
patent to the ordinary non-scientific designer, that their 
discussion, albeit important and demanding experience and 
judgment, is quite devoid of mystery. With the internal 
factors of engine-efficiency, on the other hand, although 
identified in part as early as the time of Watt and pursued 
ever since, the difficulty of prediction before construction, 
or of measurement afterward, has prevented the profession 
from yet accumulating sufficient data for a satisfactory 
science of steam-engine design. The amount of condensa- 
tion and reevaporation occurring in an engine is the re- 
sultant of a great many variables, all of which differ in 
each engine from those of any previous one with which ex- 
perience has been had. The building of a series of engines, 
in each successive one of which only one variable shall have 
been altered in comparison with the preceding one, is a 
task utterly beyond the limitations of engineering possi- 
bility, chiefly by reason of cost, but also on account of the 
indefinite length of the series needed for accomplishing a 
definite commercial result. It is therefore of the utmost 
importance to reduce the study of condensation and reevap- 
oration, and of the other points in which the actual cycle 
departs from the theoretic, to as exact an analysis of cause 
and effect as possible. It is equally important that the 
adopted method of doing this be applicable to every 
single engine-test, without necessity for using two tests 
in order to get one reading, and that a merely compara- 
tive one. 

Such an analysis is quite possible, and demands for its 
performance, besides the modicum of patient labor always 
essential to an analysis of anything, only the following 
data : — 



THE SIMPLE STEAM-ENGINE 205 

(1) The dimensions of the cylinders, including percentages 

of clearance and diameters of rods ; 

(2) The speed, which should be known to a fraction of a 

revolution per minute, by counting over a period 
of several minutes ; 

(3) The indicated horse-power, from cards taken under 

conditions, as to boiler-pressure, cut-off, etc., as 
nearly constant as. possible ; 

(4) The amount of water {steam) passing through the 

cylinder per horse-power or per tmit of time ; and 

(5) That the leakage of valves and pistons has been brought 

to the minimum possible. 

These points are all known as the result of an ordinary 
engine-test. 

At the start the following distinctions should be gotten 
clearly in mind. 

Cylindrus and Clearance. — At every stroke of the piston 
a certain volume is displaced, equal to the effective area of 
the piston times the length of the stroke. This effective 
area may be the area of the piston itself, or there may be 
the cross-sectional area of a piston-rod, a tail-rod or a trunk 
to be deducted. The volume thus displaced is known as 
the cylindrus. 

At the completion of each stroke there remains in the 
cylinder, beyond the piston in the direction of its late 
motion and between the piston and cylinder-head, a 
certain undisplaced volume, which includes all ports and 
passages back to the lips of the valves which enclose 
this space. This volume is known as the clearance. It is 
always measured and expressed as a percentage of the 
cylindrus. 

The total volume in the cylinder at any point of the 
stroke always consists of the sum of the clearance and a 
portion of the cylindrus. 



206 THE THERMODYNAMICS OF HEAT-ENGINES 

Cylinder-feed and Cushion-steam. — At each stroke of 
the engine there enters the cylinder a certain quantity of 
steam from the boiler. This steam may go to fill the 
clearance-space, it may condense in warming the cylinder- 
walls, it may go to propel the piston. Whatever its desti- 
nation it has to be supplied by the boiler. It is known 
as the cylinder-feed. It is always measured in pounds 
avoirdupois. 

At the beginning of each stroke, when the admission- 
valve opens, the entering steam finds the cylinder already 
occupied by a certain quantity of steam. This is the steam 
which has been imprisoned in the end of the cylinder by 
the closing of the exhaust-valve before the previous ex- 
haust-stroke was completed. This steam is known as the 
cushion-steam, from the frequent reliance upon its elastic 
compressibility for arresting the motion of the piston at 
the end of the stroke without undue shock upon the crank 
and connecting-rod. It is always measured in terms of the 
weight of cylinder-feed passing through the engine at each 
stroke, and is stated as a proportion of one pound of the 
latter, as, for instance : " Twenty-three hundredths of a 
pound per pound of cylinder-feed." This ratio may be 
determined on the basis of a single stroke, or of a unit of 
time. It is usually easiest to measure the cushion-steam 
for a single stroke and the cylinder-feed for an hour ; the 
two are then easily reduced to the same terms by means of 
the known engine-speed. 

This cushion-steam should always be considered as a 
part of the engine's mechanism. If the cylinder-walls and 
the cylinder-feed were to refrain from thermal interference 
with it, it would give out in expansion during the working- 
stroke of the engine just as much work as it absorbed 
during compression. In this respect it would be the exact 
equivalent of a steel spring of perfect resilience. Nor does 



THE SIMPLE STEAM-ENGINE 207 

this statement ignore the widely discussed question of the 
value of clearance and cushioning. The truth of it is not 
affected by the fact that the piston may not receive the 
work thus returned during expansion, because the exhaust- 
valve may be open or because the cylinder-walls never do 
refrain from thermal interference. Nor by the fact that, 
in some types of engines, the point in the stroke at which 
the exhaust-valve closes varies under varying load, and so 
varies the amount of steam trapped in the cylinder from 
moment to moment. All of these questions are quite sec- 
ondary and incidental to the main one, viz. : That in every 
given stroke, or complete cycle, of an engine there is a 
portion of the steam present which does not enter with 
the entering steam, which does not leave with the exhaust- 
steam, and which cannot be regarded as a means of heat- 
supply to the engine. Moreover, this cushion-steam, unless 
it can be imagined as capable of storing up or supplying 
heat indefinitely, must have had balanced up to a net zero 
for it by the cylinder-feed, by the end of each cycle, all the 
losses or gains of heat whatever which it may have made 
at earlier points of the cycle. This must hold true whether 
the gains or losses went to the cylinder-feed itself, to the 
outside air, to work, or to aught else. Such is the clearing- 
house character of the cushion-steam. 

For the cylinder-feed alone can bring heat into the en- 
gine. It alone, therefore, is chargeable with all losses of 
heat from the engine, whether they took place from it 
directly or indirectly, whether they be desirable losses (as 
in the form of work) or undesirable ones. 

It is to be repeated, then, that the first object of a rational 
analysis of efficiency is to charge up all losses against the 
cylinder-feed. This can be done only through a knowledge 
of its volume and pressure at each instant of its existence 
in the engine. If, however,. we turn to the steam-engine 



208 THE THERMODYNAMICS OF HEAT-ENGINES 

indicator as a source of light upon this question, we are 
confronted with the immediate difficulty : The indicator 
reveals volumes and pressures of cylindrus, not of cylinder- 
feed. It is the first duty, then, to translate one into the 
other. 

Let v equal the total volume existing within the cylinder 
at any point of the piston's motion. It is given by the 
abscissas of the indicator-card, when measured from a 
vertical axis placed at the proper distance from the 
end to represent the percentage of clearance. 
Let c equal the clearance-volume, a constant. 
Let z equal the portion of the cylindrus which has been 
developed at the same point of the piston's motion. 
It is measured by the abscissae of the indicator-card. 
Let y equal the volume of cushion-steam at the same 

instant. 
Let x equal the volume of cylinder-feed at the same 

instant. 
Then v = c + z, ( I01 ) 

and v=y + x. i 1 ® 2 ) 

Whence x = z -\- c — y. (103) 

In other words, from the known variable volume of cylin- 
drus, z, the unknown variable volume of cylinder-feed, x, 
may be known by first adding the constant volume of 
clearance, c, and then subtracting the variable volume of 
cushion-steam, y. 

Pressures, since all the volumes under discussion are 
freely open to another, must be the same in all the por- 
tions of steam at any given instant, and may be read from 
the indicator without translation. 

Further consideration of the above proposition regarding 
volumes raises the following question : Since the propor- 
tion of cushion-steam to cylinder-feed is the same through- 



THE SIMPLE STEAM-ENGINE 209 

out a given stroke, and since these two quantities of steam 
expand and compress together, in intimate mixture one 
with the other, why must not the volume of each vary in 
equal proportion as the pressure changes ? Physically this 
must be the case. The entire body of steam within the 
cylinder must be in practically the same condition, except 
under unusual disturbances. But analytically and mathe- 
matically a marked distinction must be drawn. Since the 
cushion-steam cannot of itself constitute a source of heat, 
any loss of heat which it experiences at any given moment 
must be imagined as immediately made good by an equiv- 
alent supply of heat to it from the cylinder-feed. Con- 
versely, any gain of heat which it may experience must be 
regarded as immediately turned over to the cylinder-feed, 
for the latter alone can carry heat out of the engine. In 
other words, the cushion-steam must always be regarded 
as expanding and compressing adiabatically . All losses 
and gains of heat, although actually experienced by either 
or both of the two steam-quantities, are to be charged 
against the cylinder-feed alone. This gives the volume of 
the cushion-steam as a known variable at any point, de- 
ducible from the known pressure at that point. This 
makes all of the terms of the right-hand member of Equa- 
tion 103 known quantities. 

The process indicated in Equation 103 is most easily 
performed graphically, especially as the quantities deter- 
mined by the indicator come to us as graphical measure- 
ments. To facilitate this work, together with some other 
mathematical deductions needing to be drawn, the author 
has projected a large blank diagram fit for rapid work upon 
the drafting-board upon a scale so large as to minimize both 
graphical error and strain upon the eye. 1 There is nothing 
about it, however, which cannot be readily produced by 

1 Published by Progressive Age Pub. Co., 280 Broadway, New York City. 



210 



THE THERMODYNAMICS OF HEAT-ENGINES 



oneself with the aid of a steam-table, though at considera- 
ble cost of time and trouble. Its essentials are represented 
diagram matically in Fig. 39. 

The sheet is divided into four portions, or quadrants, by- 
one vertical and one horizontal axis meeting in the centre 
of the sheet at O. The right-hand horizontal axis, from O 
as a zero-point, measures volumes, the scale being plainly 




n p 



Fig. 39 



indicated. The lower vertical axis similarly measures 
pressures, except that the zero-point is at the edge of the 
paper. The left-hand horizontal axis measures tempera- 
tures, the zero-point being at O, which represents the arbi- 
trary zero of the steam-tables, the melting-point of ice, or 
32 F. The upper vertical axis measures entropies from a 
zero-point at O, which also assumes the arbitrary zero of 
the steam-tables. 

From this arrangement it is obvious that the lower right- 



THE SIMPLE STEAM-ENGINE 211 

hand quadrant may be used to indicate the relations between 
pressure and volume, as upon the ordinary indicator-card. 
The lower left-hand quadrant will display relations between 
temperature and pressure ; it is also used, however, as a 
prolongation of the pressure-volume quadrant into a field 
of negative volumes, which are measured horizontally to 
the left from O on the same scale as positive volumes are 
measured from O to the right. The upper right-hand 
quadrant will display relations between volume and 
entropy. The upper left-hand quadrant will display re- 
lations between temperature and entropy, but in a position 
in reference to the eye rotated ninety degrees from that 
adopted in the previous chapters. 

It is in the premises of the diagram that these several 
axes and fields of coordinates are to be used to reveal the 
condition of the cylinder-feed within the engine during any 
given stroke. It is supposed that from the data listed on 
page 205 (this chapter) the cylindrus and the clearance 
have been calculated in cubic feet, that the weight of steam 
passing through the engine in pounds per hour is known, 
and that the engine has run under conditions so nearly 
constant that either a selected actual indicator-card, or an 
artificial one constructed from the actual ones on a wise sys- 
tem of averages, may be taken as accurately representing 
the change through which the cylinder-steam passes as the 
stroke progresses. Further, as a matter of convenience 
solely, the unit of time upon which all calculations are to 
be based is that required for the passage of one pound of 
cylinder-feed through the engine. But this unit of time is 
most conveniently expressed in number of engine-strokes 
rather than in number of seconds, minutes, etc. Any 
other unit of time, such as the hour, minute, second, or the 
single stroke, might be used with equal propriety, but at 
much greater labor of calculation. To determine this unit 



212 THE THERMODYNAMICS OF HEAT-ENGINES 

of time the number of working-strokes per hour need 
merely be divided by the number of pounds of cylinder- 
feed per hour. It is obvious that the result will depend 
largely upon the size of the engine, amongst other factors. 
It may be almost any figure, from a quite large number 
for very small engines to a fraction of unity for very large 
ones. 

The volume-scale of the indicator-card which fits prop- 
erly with the above assumptions is given by the fact that 
the total length of the card must represent the cylindrus- 
volume evolved during the chosen unit of time. This is 
plainly the actual cylindrus multiplied by the basic number 
of strokes. This is the basic volumetric scale of the in- 
dicator-card, and all other measurements taken from the 
card must be read to this same scale} 

Next, let the clearance, known as a percentage of the 
cylindrus, be laid off from the admission-end of the indica- 
tor-card as this same percentage of its total length, and 
through the point thus obtained let a vertical zero-axis of 
volumes be raised normal to the atmospheric line. If now 
any point of the card be measured as to horizontal distance 
from this axis, on the proper scale already obtained, the 
reading will be cubic feet of total cylifider-volume per pound 
of cylinder-feed passing the engine. 

In order to know the volumetric condition of the cylinder- 
feed, it now remains to subtract the variable volume of the 
cushion-steam. One of the values of this variable may be 
known from the indicator-card ; the others must be deduced 
from the first. 

1 This ratio or proportion — for it could hardly be called a scale — may 
be reduced to a comparatively convenient scale by measuring the length of the 
indicator-card in sixteenths, thirty-seconds, or hundredths of an inch, or in 
millimeters, according to what scales are at hand, and then dividing the above 
total cylindrus-volume per unit of time by the same. This gives a scale read- 
ing "so many cubic feet per sixteenth inch, or per millimeter," etc. 



THE SIMPLE STEAM-ENGINE 213 

Thiis, it is known empirically from steam-engine prac- 
tice that, except in extreme cases only, reevaporation 
is always completed early in the exhaust-stroke. This 
insures that the steam in the cylinder at the beginning of 
compression is dry saturated steam, and hence its volume 
and pressure determine its quantity. Even in steam- 
jacketed cylinders, where the tendency would be to super- 
heat the steam in the cylinder immediately reevaporation 
were complete, the heat-transfer to the dry steam is so 
slow that the departure of the truth from this assumption 
is insignificant. 

Immediately the exhaust-valve is closed, however, two 
forces begin to alter the condition of the steam : 

(1) The thermodynamic supply of energy to the steam 
by compression tends to superheat it (see page 92, point e). 

(2) The rise of temperature due to compression tends 
to conduct heat from steam to cylinder-walls, which now 
begin to present a very large surface to the comparatively 
small volume of steam. 

In nearly every actual case the second process is much 
the stronger of the two, and the cushion-steam steadily 
becomes more moist as compression proceeds. It is not 




Fig. 40 

uncommon to find an engine in which this is so far true 
that the compression finally attains a degree where the 
small plate-like volume of cushion-steam loses its heat so 
rapidly to the enveloping piston-face and cylinder-cover 



214 THE THERMODYNAMICS OF HEAT-ENGINES 

that it condenses into water isothermally as the piston 
completes the last portion of its stroke. The indicator- 
card in such cases shows a compression-curve similar to 
Fig. 40, the isothermal condensation showing at A. It is 
therefore probable that all the steam in the cylinder is dry 
saturated immediately after exhaust-closure, and even in 
those cases where a light superheat is suspected the sec- 
ond of the above two forces operates to negative this 
source of error. 1 If, therefore, a point on the card, such 

1 As this assumption of a condition of dry saturation at this point is a basic 
one, affecting all the subsequent work, it is proper to investigate not only its 
truth, which is difficult to do, but also the proportionate effect of its error upon 
the final results. To do this let it be assumed that the pressure-range of com- 
pression in the actual engine seldom exceeds that from 16 to 100 pounds 
absolute in a simple non- condensing engine, or an equivalent ratio for con- 
densing engines or for cylinders exhausting into receivers. Under such 
conditions, therefore, let the resultant final volume be compared under four 
suppositions as to initial condition: (1) Dry saturation; (2) 10% wet; 
(3) 20% wet; (4) 50 of superheat. Any engine which did not fall within 
these limits would be so abnormal in all of its activities as to either be un- 
worthy of a test at all or else worthy of especial investigation. 

The initial volumes of these four conditions may be stated relatively by 
calling the first 100. Then (2) must be 90.68; (3) 81.36; and (4) about 
107.39. After adiabatic compression to 100 pounds absolute the final volumes 
of the three would be : (1) 24.02; (2) 18.04; (3) 15-67; (4) 25.78. 

If it be assumed that the reduction of volume for all conditions is the same 
as that for dry saturation at the start, the four results are : (1) 24.02; (2) 21.78; 

(3) I 9-53; (4) 25.80. Numbers (2) and (3) are over 20% too large; (4) is 
practically correct. If a similar assumption be applied to steam of original 
condition of (2), the four- results are: (1) 19.85; (2) 18.04; (3) 16.14; 

(4) 21.34. Now (1) and (4) are some 17% too small; (3) is within 3% of 
correct. 

When it is considered that these assumed conditions are extreme and that 
the weight of cushion-steam rarely exceeds one-fourth of the cylinder-feed 
(which would reduce these percentages of error, when compared with the 
whole quantity of steam in the cylinder, to one-fifth of the figures stated), 
good basis is seen for the conclusions: (1) An assumption of dry saturation at 
the beginning of compression leads to no appreciable error in results if the steam 
be actually dry or superheated at start, and to an error of from 1 % to 5 % if 



THE SIMPLE STEAM-ENGINE 215 

as B, be chosen, which represents an instant when the 
exhaust-valve is known to have been only recently com- 
pletely closed, and if the volume and pressure at B be 
measured by the scales already described, they will reveal 
the quantity of cushion-steam present at the instant per one 
pound of cylinder-feed. 

Let these measurements be laid off on the diagram at 
the point Y, Fig. 39, the volume being regarded as a nega- 
tive one. Through Flet an adiabatic AB be constructed. 
It will represent the volumes which the cushion-steam 
ought to occupy for every given pressure in the cylinder. 

In the published blank diagram will be found, in this 
portion of lower left-hand quadrant, a number of such adia- 
batics. With them as guides, when once the point Y is 
determined, the desired adiabatic may be laid off by eye. 
These adiabatics are the pressure-volume ones, for the 
several weights noted on the curves themselves, correspond- 
ing to the single entropy-temperature adiabatic shown 
in the entropy-temperature quadrant. All agree in being 
based upon an original dry-steam point at an absolute 
pressure of 41.56 pounds per square inch, at which 
pressure the volume of one pound is 10 cubic feet. 
Should the original observed point Y happen to fall at this 
pressure, the weight of cushion-steam present may be read 
directly from the weights pointed on the several curves. 
If not, however, as would usually be the case, the weight 
as given by the curves need only be multiplied by the 
percentage indicated in the list of them to be found in the 
extreme lower left-hand corner of the sheet, the proper one 

the steam be quite wet at the start; (2) Assumption of moisture at the start 
leads to no appreciable error if the steam actually be as moist as assumed or 
even much more so, and to an error of from I % to 3 % if it actually be drier 
than assumed. The actual error, under average conditions, will not exceed 
one-third of these figures. 



2l6 THE THERMODYNAMICS OF HEAT-ENGINES 

being chosen according to the pressure-level of the point 
Y, in order to give the actual weight of cushion-steam 
present per pound of cylinder-feed. A knowledge of this 
weight is not needed, however, for any further process of 
this analysis, but may be desired for other reasons. 

In the straight vertical axis which was drawn outside the 
clearance on the indicator-card is to be had a zero-axis of 
total cylinder-volumes, as already noted. The object in 
view is to deduct from those volumes the variable volumes 
of cushion-steam in order to find the variable volumes of 
cylinder-feed. If, now, the line AB, Fig. 39, be used as a 
curvilinear zero-axis of total cylinder-volumes, and if from 
it any measurement taken from the indicator-card, such as 
SZ (translated into the proper scale), be laid off to the 
right, then the axis OP must automatically cut off from it 
the volume SR which should be properly occupied by the 
cushion-steam at this pressure. The remainder RZ meas- 
ures the volume which the cylinder-feed would have were 
all the heat-losses and gains charged against it. OP there- 
fore becomes the zero-axis of cylinder-feed volumes, and the 
lower right-hand quadrant becomes a rectangular coordinate 
field revealing the pressure-volume conditions of one pound 
of cylinder-feed. q.e.f. 

The indicator-card should now be reproduced, point by 
point, in the above manner. It will appear, distorted in 
form, something like the curve EFZGHIK, Fig. 39. (By 
handling the points in horizontal pairs, as Z and Z ,u ', much 
labor can be saved.) In this form it is of no immediate 
value, but from it can easily be deduced the entropy-tem- 
perature changes incurred by the cylinder-feed, and they, 
in turn, will show plainly just when and in what amount 
heat was lost or gained. 

Let attention be confined to the illustrative point Z. 

So long as the distorted indicator-card lies between the 



THE SIMPLE STEAM-ENGINE 2\J 

curves AB and MN the assurance is had that the cylinder 
is filled with saturated steam more or less moist; for the 
curve MN is drawn as a "saturation-curve," representing 
the relation between volume and pressure of dry saturated 
steam ; or, in other words, MN is one of the columns of 
the steam-table represented graphically. In the tempera- 
ture-pressure quadrant is a similar curve, mil, showing the 
relation between the temperature and pressure of dry satu- 
rated steam. Therefore if a line be run horizontally across 
from Z until it meets mn, at L, and thence a vertical line 
be run to R" and Q", it is certain that the entropy-tempera- 
ture condition of the steam at the point Z must be shown 
at some point, such as Z' { ', on the line R" Q" . Which is the 
proper point, however, can be known only through informa- 
tion as to volumes. 

If a horizontal be run from R" to the vertical axis at R' , 
the latter point must represent the relation of entropy to 
volume for water at the pressure shown by Z\ for the 
volume of one pound of water is so small as to be invisible 
on the scale chosen. The volume of one pound of dry 
steam, however, of the pressure shown at Z is shown by 
the point Q, and may be carried up into the entropy-volume 
quadrant by the ordinate QQ' . The entropy of one pound 
of dry steam of the pressure and temperature shown by Z 
is shown at Q", and may be carried over into the entropy- 
volume quadrant by the abscissa Q" Q! . The intersection 
of QQ' and Q" Q' at Q' gives the latter point as a measure 
of the relation of entropy to volume for one pound of dry 
steam at the pressure shown by the point Z. 

Referring to the Equation 21, page 87, it will be remem- 
bered that when a given weight of water changes to steam 
by the addition of heat under constant pressure, the incre- 
ments of heat, entropy, volume, and weight of vapor 
formed were all proportioned. Hence the relation of 



2l8 THE THERMODYNAMICS OF HEAT-ENGINES 

entropy to volume for a steam-and-water mixture of a 
given pressure must be represented by a straight line. In 
the case under discussion the points R' and Q' must be the 
extremities of such a line. Joining them gives a line to 
which the volume of any such mixture as RZ may be 
referred, by means of the ordinate ZZ' , to find its en- 
tropy OZ"" . Carrying across the horizontal Z'Z" until it 
intersects R"Q", must determine the point on the entropy- 
temperature quadrant which illustrates the thermal con- 
dition of the cylinder-feed at the point Z. 

It will be noted that while there is only one temperature- 
pressure saturation-curve, mn, for all weights of steam and 
all conditions, and while there is only one pressure-volume 
saturation-curve, MN, for a given weight of steam, there 
are any number of entropy-volume curves similar to R'Q' 
for a given weight of steam-and-water mixture, each serv- 
ing for one particular pressure or temperature of mixture 
only. 

The successive transfer of all points of the distorted 
indicator-card in the manner just described results in an 
entropy-temperature diagram of the actual cycle under- 
gone by the cylinder-feed in the actual engine. This dia- 
gram, rotated through ninety degrees to bring it into the 
position in which these diagrams were always depicted 
in the first part of the book, would have something the 
appearance of Fig. 41, although wide variations are to be 
expected from engines of different types and sizes. 

This diagram depicts the actual thermodynamic pro- 
cesses of the cycle performed within the cylinder. In it 
may be seen just when, how, and to what extent heat- 
energy reaches the steam and work departs from it. It 
shows the advent of heat by the increase in entropy 
wherever the curve takes a right-hand departure. It 



THE SIMPLE STEAM-ENGINE 



219 



shows, in literal fashion, just when this entropy falls down- 
temperature, doing work, the path by which it falls, and 
the amount of work done in the fall. It shows when and 
how the fallen entropy leaves the engine and by what path 
the unexpelled remnant is again raised to the head-level 
of temperature. It shows how much temperature-fall is 
wasted in getting the heat into the engine and how much 
in getting it out. It shows the leaks of heat into and out 
of the steam as it rises and falls, doing work, and just how 
much energy-waste is represented in these leaks. 




Fig. 41 



It is important to study this photograph of the cycle 
carefully and to separate by accurate analysis these vari- 
ous factors. 

It is first necessary to refer this actual cycle to the one 
which would have been followed by a perfect engine work- 
ing within the same opportunities and limitations. Since 
the engine is a steam-engine, the utmost that can be 
expected of it is the Rankine cycle. If the engine were 
supplied with dry saturated steam, this cycle would be 
shown by the lines ABCD; if with wet or superheated 
steam, this cycle would be modified as illustrated in Prob- 
lems 14 and 17. 



220 THE THERMODYNAMICS OF HEAT-ENGINES 

It is a matter of less certainty, however, as to just what 
should be the level of the limits at top and bottom. If the 
engine be a simple, non-condensing one, situated near its 
boiler and with ample pipe-connection between, BC would 
clearly represent boiler-temperature and AD that corre- 
sponding to the barometric pressure of the atmosphere. 
Should there be a long or insufficient pipe-line feeding the 
engine, however, it is important to distinguish between 
boiler-pressure and throttle-pressure. If it is desired to 
include and examine the evil effect of the pipe-line upon 
engine-efficiency, both temperature-levels may be drawn in ; 
and while the question of piping-losses involves many other 
factors than the mere thermodynamic effect upon the 
engine, this last would be at least clearly and accurately 
represented, both absolutely and as to proportion to other 
losses, by this graphic comparison. 

If the engine be a condensing one, the problem is still 
more open to discussion. The simplest and, when the 
engine is considered alone, the most accurate plan is to 
draw AD at the temperature-level corresponding to the 
actual vacuum. When the efficiency of the condenser and 
air-pump is to be included, however, as if this apparatus 
were a part of the engine whose efficiency is being ana- 
lyzed, AD should be drawn at the maximum, or even the 
mean, temperature of the condensing water. 

When the engine is a multicylinder compound, the result 
will appear as a vertical series of such cycle-diagrams, one 
over the other. The gaps between represent receiver- 
losses. In such cases it is usually sufficient to draw be- 
tween the cycles horizontal lines at the temperatures of 
saturation at the recorded receiver-pressures, trusting to 
engineering judgment to draw the proper deductions from 
the case without attempt at greater refinement of analysis 
on the graphical diagram. The question as to where the 



THE SIMPLE STEAM-ENGINE 221 

adiabatic CD should be drawn is less simple, however, and 
will be taken up later as a separate topic. 

Assuming that we are concerned with a single cylinder 
only, supplied with dry steam, the following conclusions 
are safe. 

Wire-drawing. — When the engine took steam, it should 
have done so along BC, cut-off occurring at C. Instead, 
it does so along BE, cut-off occurring at E. The energy 
available for doing work at E is less than that at C by two 
quantities, viz. : (i) The triangle BEE, (2) the rectangle 
FCDG. The second quantity of heat lost is measured by 
the rectangle between EC and the axis of absolute zero of 
temperature, not shown on the diagram but easily drawn 
in, or it may be obtained as the result of a single multipli- 
cation ; but only FCGD is available for doing work. 

The first of these is plainly due to wire-drawing, for wire- 
drawing cannot rob steam of heat ; it merely reduces the 
temperature. The loss of availability for work due to loss 
of temperature without loss of heat is shown by the area 
BEE, and this area is therefore chargeable against wire- 
drawing, as a loss due to it alone. 

Initial Condensation. — The second loss, however, is due 
to loss of entropy, and not of temperature, and must there- 
fore be due to an abstraction of heat. As the only known 
process present which can abstract heat is condensation on 
the cold cylinder-walls, this rectangular loss is chargeable 
against initial condensation. 

Condensation during Expansion. — The energy available 
at cut-off at E would indicate that a perfect engine, by 
following the adiabatic EC, might develop the cycle 
ABEGA. But instead of doing this the actual engine 
follows the path EH. Its departure to the left indicates 
abstraction of heat by continued condensation. Some of 
this heat, however, by falling from later points of the path 



222 THE THERMODYNAMICS OF HEAT-ENGINES 

BE to points vertically below in EH, has performed some 
work. The loss due to this further condensation is there- 
fore only the area between EH and KG. 

Reevaporation. — Again, the engine, instead of expand- 
ing the steam adiabatically after condensation is complete, 
in which case it would follow the path HK, shows a devia- 
tion to the right along HL. This shows the addition of 
heat to the steam. This heat can come from no other 
source than the cylinder-walls, and must be due to the fact 
that the temperature of the steam has fallen by expansion 
below that of the walls, whereas originally it was above it. 
The addition of heat must result in the reevaporation of 
some of the moisture on the cylinder-walls. The steam- 
heat thus brought in is of course available for doing work, 
but by no means to the same extent as the original heat, 
which came in at maximum temperature. The work which 
it can do must be measured by the area beneath HL, or 
HLMK, and this area may therefore be credited as a gain 
due to reevaporation. 

Incomplete Expansion follows next in the list of devia- 
tions from the adiabatic path to the lower level. Instead 
of following the path LMA back to A the temperature-fall 
next departs to the left, when the exhaust-valve opens, 
along LN, and, if it were not arrested by a still further 
impediment to perfect action, would reach the A-\evel at Q. 
The triangular area LQM is therefore chargeable to ex- 
haust before expansion is complete. It is always to be 
remembered, however, that for practical reasons which will 
be explained later complete expansion is never profitable. 

This line Z<2,'when compared with a theoretic constant- 
volume curve through L, gives the exaggeration of the loss 
due to incomplete expansion by clearance. This consider- 
ation, with those discussed on pages 223-225, gives an 
absolute measure of the deleterious effect of clearance. 



THE SIMPLE STEAM-ENGINE 223 

Back-pressure. — It next develops that the cycle never 
attains the lowest possible temperature-limit, DA. It never 
falls below JVR. The area representing the resultant loss, 
NQAR, usually measures a nominal rather than a real loss. 
Some back-pressure is inevitable. To supply ports and 
valves of the enormous size necessary for eliminating it 
would be very foolish. As with the loss due to incomplete 
expansion, the only question is whether it be excessive or not. 

Cushion and Clearance. — With the next deviation from 
the theoretic, that of the path STUB from the water-curve 
RB, arises a much more pregnant question of steam-engine 
design. This question lies between two broad types of 
steam-engine : (i) Those designed to have a minimum pos- 
sible clearance, with little or no cushion, and (2) those de- 
signed to have a liberal clearance, which is to be filled with 
cushion-steam at each stroke compressed to as nearly boiler- 
pressure as is practicable. The first is naturally the drop- 
cut-off, slow-speed type ; the second is the positive-gear, 
high-speed type. Before discussing the question, however, 
for it is a lengthy one, it were well to decide just how the 
amount of heat-work lost during compression is to be 
measured from the actual cycle before us. 

It is obvious that at the point 5 the cylinder-feed pos- 
sesses more heat than it theoretically should, and that 
during compression heat to the amount STR is abstracted. 
This little triangular area reveals the departure from the 
cylinder of exhaust-heat along the line ST which should 
have gone out by the path SR. 

At some point in the compression, however, as at U, the 
cycle shows the cylinder-feed as having less heat than it 
would if it were water of the given temperature, which is 
manifestly impossible. What has happened is plainly that 
the cushion-steam, which alone occupies the cylinder in the 
actual engine, although theoretically all of the cylinder-feed 



224 THE THERMODYNAMICS OF HEAT-ENGINES 

is present in the form of water, has lost some heat to the 
walls during its compression. The mathematical elimina- 
tion of the cushion-steam, however, causes this loss to 
appear as one drawn from the cylinder-feed. 

At first sight it might appear that what was wanted was 
an analysis of what happened to the cushion-steam during 
its compression. In reality, however, such determinations 
are misleading. For instance, it might thus be found that 
the cushion-steam at some point in its compression con- 
sisted of 50% water. This would seem a tremendous loss 
of heat, which should be remedied at any cost. If, how- 
ever, the cushion-steam amounted to only 10% of the 
cylinder-feed, the lost heat reduces to 5% in terms of cyl- 
inder-feed, or one of the smallest in the engine. And 
since it is cylinder-feed alone which is a source of expense 
to the engine-user, all losses must be reduced to that basis 
of comparison. 

It is obvious, too, that all heat lost by the cushion-steam 
during compression must be made good by the cylinder- 
feed entering at the next stroke before the latter can begin 
to accomplish any useful result. It is therefore quite 
proper to consider the point where the heat-abstraction 
from the cushion-steam reaches the extreme as the zero of 
all heat-gains and losses by the cylinder-feed. This is 
easily done graphically by drawing through the point U a 
curve parallel to BR, such as bUr, tangent to BTR. 
This arbitrarily moves all reference-points to the left by 
the distance Bb. A new limiting adiabatic must therefore 
be drawn parallel to CD and at the same distance to the 
left, as at cM. This alters the apparent value of heat lost 
in initial condensation from that prescribed on page 221. 
The other losses and gains are unaltered. What is in 
reality true is that a large portion of the initial condensa- 
tion occurring in every cylinder is due to heat-losses 



THE SIMPLE STEAM-ENGINE 225 

incurred before any steam was admitted, which losses are 
stored up by the cushion-steam for draft upon the cylinder- 
feed when the latter shall next appear in the cylinder. 
The process just defined separates these two responsibili- 
ties for condensation. The area UrSUBbU is chargeable 
against clearance and cushion-steam. The area FcMG is 
due to condensation which occurs after the clearance has 
been filled with boiler-steam and which would take place 
were the clearance zero. It is this preliminary demand for 
boiler-steam to reimburse the cushion-steam and to carry 
on the preliminary condensation which occurs before the 
stroke begins which constitutes the real reason for the lead 
of the valve. When it is remembered that the greatest 
part of the surface exposed to boiler-steam by the time cut- 
off takes place is already exposed to it when admission 
takes place into the clearance-space at dead-centre, it is 
obvious that there is good basis for expecting this prelimi- 
nary demand for steam to constitute a goodly proportion of 
the total supply. 

It is also obvious, from the foregoing argument, that it 
is the c\ea.ra.nce-szijface, rather than the clear ance-vo/ume, 
which is so deleterious to steam-engine efficiency. 

As the assumption of initial dryness on the part of the 
cushion-steam is the only assumption entering the entire 
analysis, it may be insisted that the conclusions reached by 
means of the entropy method are exact to an unusual 
degree. If, therefore, the entropy-analysis should reveal 
that the area chargeable to cushion-condensation was so 
great, in any given instance, as to overbalance any gain 
from diminished first cost, space, etc., which the cushion, 
through the medium of the high rotative speed which it 
permits, has developed, then the minimum-clearance design 
should obviously be chosen in preference for this particular 
service. 



226 THE THERMODYNAMICS OF HEAT-ENGINES 

With the foregoing means of obtaining an exact knowl- 
edge of existing facts as a basis, some very plain and sim- 
ple conclusions may be drawn concerning several features 
of steam-engine design which frequently find themselves 
under discussion. 

Size and Speed 

If by the size of an engine is meant the dimensions of its 
cylinder, then it may be stated as a well-known fact that 
the larger the cylinder, the less is the condensation, because 
the less the surface exposed per pound of cylinder-feed. 
The internal surface of similarly shaped vessels, such as 
engine-cylinders, must plainly increase directly with the 
square of any one dimension. But as the cubical contents 
at the same time increase directly with the cube of the 
same dimensions, it is plain that the ratio between cylinder- 
surface and the amount of steam contained must vary 
inversely as the first power of the same dimension ; whence, 
the larger the cylinder, the less will be the percentage of 
condensation. 

More commonly and more accurately, however, the size 
of an engine is understood to refer to its power. This 
being so, is it equally true to say that the larger the engine, 
the less the condensation ? If the variation in power be- 
tween two engines subject to comparison is gotten by vary- 
ing the cylinder-dimensions, while speed and pressure are 
kept constant, then it is plain that the proposition still 
holds true. If, however, the increased power be gotten by 
increased speed, the cylinder-dimensions being kept con- 
stant, the problem is altered. Here the same surface is 
exposed to the same fluctuations of temperature a greater 
number of times per minute. As the amount of cylinder- 
feed is also roughly proportioned to the speed, the propor- 
tion of condensation could only remain the same, as the 



THE SIMPLE STEAM-ENGINE 227 

speed increased, if the absolute quantity during each cycle 
of temperature remained constant. But as it takes time to 
transfer heat from steam to cylinder-wall and back again, 
and as the time occupied by each cycle diminishes as the 
speed increases, the absolute condensation per cycle will 
plainly not remain the same ; instead, it will decrease, and 
hence the proportion of condensation suffered by the 
cylinder-feed must be said to decrease as the speed in- 
creases. But, whereas it was quite safe to assume that in 
alterations of cylinder-dimension the amount of condensa- 
tion varied in true and simple proportion to the surface, it 
is not by any means probable that the condensation upon 
any given surface is proportional to the time of exposure. 
For the heat is not carried away by the metal, but is 
merely stored up within itself, and it is easy to imagine 
that the first heat imparted to the walls might raise its tem- 
perature sufficiently toward that of the steam so that the 
rate of heat-transmission would be substantially altered. 
Indeed, it is safe to say that the condensation must 
increase with the time of exposure in some ratio which is 
some fractional power of the time. If this should be true, 
then it would appear that it were more profitable, so far as 
condensation is concerned, to increase an engine's power 
by increasing the size of cylinder rather than by speeding 
up, and as purely mechanical argument also advises the 
running of larger cylinders at slower speeds, it is plain how 
thermodynamics and mechanics coincide in supporting the 
natural tendency visible in actual designs. 

But it is also to be noted that since increase in speed is 
actually conducive to thermodynamic efficiency, but only 
less so than increase in dimension, the running of large- 
powered engines at high speed finds no censure in thermo- 
dynamic argument, when it is warranted by mechanical 
considerations. 



228 THE THERMODYNAMICS OF HEAT-ENGINES 

Steam-pressure 

As the range of pressure, and therefore of temperature, 
in an engine increases, the theoretic efficiency plainly 
increases. At the same time, by reason of this same tem- 
perature-range, the condensation must increase. At first, 
as one starts from a very low pressure-range, the theoretic 
gain will be very great because it is almost fully propor- 
tional to the temperature-range and because the latter 
increases, in the lower field of pressures, more rapidly than 
the pressure. But as higher pressures are reached, all of 
these conditions are first annulled and then reversed, and 
the increase in condensation is found to more than offset 
the gain in heat-available-for-work. The point where bal- 
ance is established between the two gives the maximum 
pressure-range which it is profitable to impose upon the 
cylinder for purely thermodynamic considerations ; though 
for other reasons higher pressures are often found profit- 
able in special services, such as locomotives or torpedo- 
boats, where it is all-important to get a maximum of power 
out of each inch or pound of engine. 

This limit may be determined only by experiment upon 
each type of engine considered. For reasons stated in the 
preceding article, it is plain that the limit would be lower 
in small engines than in large. It is generally understood 
to lie between 90 and 120 F. of temperature-range within a 
single cylinder, the lower figure for small cylinders and the 
higher for large ones. It should also be understood, how- 
ever that when applied to unjacketed cylinders these figures 
give rather the maximum range obtainable without real detri- 
ment from condensation than the minimum range permissible 
without net loss from thermodynamic reasons ; for the natu- 
ral tendency is always to take the greatest permissible range, 
for the sake of capacity or power, rather than the least. 



THE SIMPLE STEAM-ENGINE 229 

The Steam-jacket 

It is the role of the steam-jacket to prevent or minimize 
condensation in the cylinder. Since the heat supplied to 
the steam through the medium of the steam-jacket is good 
boiler-heat, obtained at as great an expense as steam-heat 
entering by way of the throttle, some explanation is neces- 
sary as to why any gain in efficiency is possible by its aid. 
Indeed, since the steam-jacket is free to radiate heat into 
the exhaust during half the time, and since, even during 
the working-stroke, a large part of the heat supplied by it 
to the working-steam is imparted after cut-off, when the 
availability for doing work is much less than is the case 
with boiler-steam, explanation is also necessary why the 
steam-jacket should not constitute a distinct source of loss 
of efficiency. The answer to both points is found in the 
fact that the one thing in an engine-cylinder which is able 
to very much exaggerate the detrimental influence of the 
cylinder-walls is moisture. Able to exchange heat with 
the steam much more rapidly than can the iron of the 
walls, and having, too, a much higher specific heat, water 
is very much more active than metal in carrying on that 
condensation and reevaporation which has always been 
the chief bar to the attainment of maximum theoretical 
efficiency. 

It is not a sufficient guard against the phenomenon to 
provide the engine with perfectly dry steam, although the 
presence of wet steam makes the situation very much worse 
than if it were dry ; for even with dry steam the initial con- 
densation which must take place upon the walls as the 
clearance fills with boiler-steam forms a nucleus of moisture 
which invites additions in a geometrical ratio. 

The steam-jacket prevents the appreciable accumulation 
of moisture upon the internal surface of the cylinder-walls 



230 THE THERMODYNAMICS OF HEAT-ENGINES 

by warming up the cylinder with steam-heat, the resultant 
moisture from which remains outside the cylinder and is 
drained away. That is the real and essential difference 
between a jacketed and an unjacketed cylinder. The heat 
supplied from the boiler for radiation is just as great and 
just as costly in either case; but in the former the result- 
ant moisture does not remain in the cylinder as an active 
carrier of heat into the exhaust, whereas in the latter it 
does. 

There is an additional gain, too, due to the fact that the 
mean temperature of the cylinder-walls is kept nearer to 
boiler-temperature. Because the walls are open to the ex- 
haust during the back-stroke, the steam-jacket cannot quite 
bring them to boiler-temperature. But the initial conden- 
sation is much decreased in amount, and its reevaporation, 
instead of awaiting the fall of temperature within the body 
of expanding steam below a mean between boiler and 
exhaust-temperatures for its inception, begins as soon as 
cut-off takes place and usually finishes before the working- 
stroke is complete. This being so, the steam-jacket can 
lose very little heat to the comparatively dry exhaust-steam, 
and by the time cushion takes place can get the walls well 
back toward boiler-temperature. The result of its action 
therefore is : — 

(1) To much decrease the amount of boiler-heat under- 
going free fall within the cylinder by keeping out of the 
cylinder the moisture developed in heating it ; 

(2) To impart the bulk of what does undergo this free 
fall of temperature to the working-steam at an earlier point 
in the cycle, before the latter's expansion is complete, 
whereby a portion of its temperature-fall to exhaust-tem- 
perature is utilized. 

Since the prime office of the steam-jacket is to diminish 
condensation, it must be of the most value where conden- 



THE SIMPLE STEAM-ENGINE 23 I 

sation is excessive ; that is, where a high thermodynamic 
efficiency is, or is thought to be, valuable, and is sought 
through the medium of ( 1 ) a large ratio of expansion or 
(2) slow speed of rotation. Under these conditions a good 
jacket will show an increase of efficiency varying from 5 to 
15%, with probably 7% or 8% as a fair average. 

It is to be especially noted, however, that under wrong 
conditions the steam-jacket may become a source of expense 
instead of economy. The majority of engine-builders, rat- 
ing small and large together, omit it from their designs, 
and in this the majority of them are right. In debating 
the question as to when jackets are desirable and when 
they are not, the first fact to be noted is that a steam- 
jacket is of itself a very expensive and undesirable feature 
in an engine, and the saving which it develops must be a 
substantial and a certain one if the net result of its addi- 
tion to the engine is not to be a distinct loss. A jacket is 
difficult and expensive to construct. The core for it is 
large and thin. Its presence usually entails coring several 
other portions of the casting where cores would otherwise 
be unnecessary. The percentage of cylinder-castings lost 
in the foundry is very much higher with jackets than with- 
out. The joints between barrel and head of cylinder at 
either end are much complicated by its presence. If it is 
remembered that the great bulk of the condensation of the 
working-steam takes place in the clearance while the piston 
is at, or very near to, the end of its stroke, it will be plain 
that the portions of the engine which especially need to be 
jacketed are the cylinder-heads, and to connect these jackets 
with those around the barrel and to avoid too close contact 
with the stuffing-box, all involves difficulty and expense in 
construction and maintenance. Therefore, while it is un- 
doubtedly true that a steam-jacket would somewhat increase 
the thermodynamic efficiency of any cylinder whatever, it 



232 THE THERMODYNAMICS OF HEAT-ENGINES. 

is only when this increase is substantial that it may coun- 
terbalance the commercial losses involved in its adoption. 

The settlement of this question necessarily involves an 
understanding of the relation existing between thermo- 
dynamic and commercial arguments in any given case. 
But its discussion applies with equal weight to all of the 
factors which enter into engine-efficiency which have 
already been discussed : size, speed, ratio of expansion, 
clearance, and cushion. It is therefore proper to recon- 
sider them all in the light of this one argument. 

The design of a steam-engine, as is true of any merchant- 
able article, properly follows the lines upon which demand 
is expressed. The demand coming from the buyers of 
steam-engines naturally divides itself into two broad sorts, 
if minor differences due to special conditions be ignored. 
Of these, the first comes from him to whom the item of 
power is a large and important one in his business ; the 
second comes from him to whom power is a minor factor. 
Examples of the first are electric lighting, electric genera- 
tion in general, large textile mills, paper mills, marine pro- 
pulsion, etc. Examples of the second are machine-shops 
and the general run of factories of moderate size wherein, 
as in the machine-shop, the pay-roll or the cost of raw 
material is a very much larger item than is the cost of 
power. All of the former class have presumably plenty 
of capital to invest in steam-plant, for that is their business, 
and return upon that capital will be a maximum when the 
coal-consumption is the least. Of the second class, those 
who have plenty of capital invest it in acquiring the best 
plant possible of the sort which really does the earning 
of the returns from their business, and which consists of 
machine-tools, overhead cranes, and similar facilities for 
doing the best possible work in the minimum of time and 



THE SIMPLE STEAM-ENGINE 233 

with the least waste of raw material or labor. The power 
plant of such a factory is properly a secondary considera- 
tion. To it, therefore, is accorded only sufficient capital to 
prevent its being actually bad. 

In addition to this latter class, too, should be considered 
those to whom the cost of power is a prominent item in the 
cost of doing work, but who have to approach every de- 
partment of their business, owing to insufficient capital, 
in the attitude of cutting down investment to the lowest 
possible point without regard, always, to whether further 
investment might not be profitable. 

These two sorts of buyers therefore demand, the one an 
engine which will operate upon the highest fuel-efficiency 
without regard to first cost, the other an engine which will 
bring the first cost for power and its incidentals to the 
lowest possible figure without regard to more than a mod- 
erate degree of efficiency. It so happens that these two 
attitudes lead to decisions in regard to the several points 
of design just discussed which are singularly harmonious 
in their combination into a complete type of engine. These 
points are : — 

1. Degree of expansion, including the question of com- 
pounding. 

2. Valve-motion. 5. Speed. 

3. Clearance. 6. Steam-jacketing. 

4. Cushion. 7. Condensation. 

1. Degree of expansion. — It is plain that the greater the 
ratio of expansion, the greater is the ideal efficiency. It is 
also obvious that the greater the degree of expansion, the 
greater are, not only the thermodynamic losses due to cyl- 
inder-condensation, etc., but also all those expenses due to 
larger cylinders and earlier cut-offs as well ; for the entire 
cost of an engine-plant is closely proportional to the size of 



234 THE THERMODYNAMICS OF HEAT-ENGINES 

cylinder. A comparatively late cut-off may be easily gotten 
by means of a single valve and a very simple valve-motion ; 
but to attain an early cut-off, and at the same time preserve 
the proper distribution of exhaust and compression, de- 
mands the use of from two to four separate valves, and 
often more than one eccentric. Therefore it can be safely 
said that there is one broad type of engine which carries 
the attainment of efficiency by means of higher ratios of 
expansion at the expense of larger cylinders and more 
complicated valve-gear; there is another which foregoes 
this efficiency to an appreciable degree, to the end that 
cylinders may be small and the valve-gear simple. In this 
connection it ought also to be remembered that large cyl- 
inders mean larger engines in every respect, and larger 
engines mean heavier and more costly foundations, more 
care, more oil, more space about them, heavier cranes over- 
head, etc., etc. In fact, it may be safely said that the 
entire cost of construction of engine-room and its mainte- 
nance, aside from the cost of fuel, bears a rough but fairly 
accurate proportion to the size of cylinder employed. 

The foregoing remarks apply equally to the question of 
the degree of completeness of expansion, or of terminal 
pressure, which is somewhat independent of the question 
of ratio of expansion. It is a very common fallacy that 
the indicator-card with the sharpest toe, other things being 
equal, shows the best efficiency. So false is this that it 
may be broadly and responsibly stated that any indicator- 
card with a sharp toe is inherently bad. The justification 
of this statement lies in the following argument : It is 
obvious that of the pressure in the cylinder at any moment 
only that exerting a pull upon the belt is of any value to 
the engine-owner ; the rest is expended in merely keeping 
the engine in motion. In other words, if we translate 
engine-friction into its equivalent in mean pressure upon 



THE SIMPLE STEAM-ENGINE 235 

the piston, we shall find the equivalent to amount to from 
about 4 pounds per square inch for large low-pressure 
cylinders to about 15 pounds for high-pressure cylinders. 
Whenever the pressure upon the piston is less than this, 
the belt must be dragging the piston after it, and the net 
result is a distinct loss. Moreover, since every inch of 
cylinder costs money to build, to erect, to maintain, and to 
operate it, the engine is a distinct source of loss until it is 
developing enough power of sufficient net value to make 
good this loss. If this power be also translated into mean 
effective pressure and added to the former figure, the re- 
sults will vary between the extreme limits of about 5 and 
25 pounds, or ordinarily between 7 and 20 pounds, per 
square inch of mean effective pressure. Of these figures 
the smaller apply to low-pressure, the larger to high-press- 
ure cylinders. If the terminal pressure in the cylinder be 
allowed to fall below these figures, it is proof that the engine 
is not developing and delivering to the belt enough money- 
making power to fully pay for its keep. The plant would 
be more efficient under a later cut-off, either in a smaller 
cylinder or in the same cylinder under lower initial 
pressure. 

The wide prevalence of the fallacy referred to in this 
connection is almost wholly due to the absurd practice of 
uniformly making guarantees and reports of engine-effi- 
ciencies upon the basis of the indicated instead of upon 
the brake horse-power. Deception as to what is a profit- 
able degree of expansion is only one of the many ways in 
which the engineering public loses from this habit. 

The question of the profit of compounding or not comes 
under exactly the same argument. The details of the sci- 
ence of compounding will be treated later under the proper 
heading ; but it is important to note that the commercial 
efficiency of a steam-plant is not necessarily improved by 



236 THE THERMODYNAMICS OF HEAT-ENGINES 

compounding, and that many existing compound engines 
are operated at a loss compared with what might be accom- 
plished by a single cylinder. But in the great majority of 
cases the mistake is in the other direction. Single cylin- 
ders are adhered to for the sake of cheapness when the 
additional money needed for compounding would earn a far 
greater return than any other money invested in the business. 
It might also be misapprehended from the above that 
compounding should only be regarded as an exaggeration 
of the policy of the short cut-off, four-valve type of engine, 
and to be adopted only after passing through this prelimi- 
nary degree of search after efficiency. This is not true, 
however. There are many mechanical and commercial 
advantages to be conserved by retaining the compact, 
single-valve, high-speed type of machine, even when search 
ing after greater efficiency by means of compounding. 
But the resultant type is necessarily an intermediate and 
more or less special one, limited properly to a compara- 
tively narrow, although absolutely a very large, field of 
sizes and of diversity of employment. 

2. Valve-motion. — This point has already been partly 
covered. The valves and gear fit to produce early cut-offs 
are comparatively expensive to build and to maintain. The 
question is also intimately connected with that of 

3. Clearance. — It happens that the valves fit for pro- 
ducing a very short and sharp cut-off also permit the 
attainment of a very small clearance in the cylinder. As 
clearance has been shown to be always somewhat detri- 
mental to efficiency, here again the coincidence is harmo- 
nious with the other points in dictating large cylinders, 
four valves, and small clearance as characteristic of the 
more efficient type. It also happens that the best forms 
of single-valve engines always involve large cylinder-clear- 
ance. This brings into the discussion the next point : — 



THE SIMPLE STEAM-ENGINE 237 

4. Cushion. — The detrimental effects of clearance can 
be very largely, although not entirely, counteracted by the 
use of sufficient cushion to bring the pressure in the clear- 
ance approximately to boiler-pressure by the time dead- 
centre is reached. It also happens that when a single 
valve and eccentric are used, the attainment of a cut-off 
earlier than about three-quarters stroke is impossible with- 
out making the exhaust-closure also correspondingly early. 
So that it appears that in the type of engine which needs 
cushion to make it efficient considerable cushion is mechan- 
ically inevitable. It is a fortunate coincidence that in this 
type of engine a great deal of cushion may be had very 
easily. 

5. Speed. — It was pointed out in the analysis of the 
causes of cylinder-condensation that, for an engine of any 
given power, that phenomenon was probably not at all 
affected by speed. But the many items of cost of opera- 
tion of an engine due to causes other than cylinder-conden- 
sation are all very sharply affected by speed. The higher 
the speed, the smaller and cheaper are the cylinder, the 
foundations, and the engine-room, the lighter are the belts 
and shafting, and the better the regulation. That there 
are any mechanical disadvantages or costs to be incurred, 
under proper design, from high engine-speed is not to be 
urged to-day. Twenty years ago high speed had to apolo- 
gize and fight for itself, as a mechanical monstrosity. To- 
day even the advocates of the slowest speeds have so 
increased their rates of rotation that it is generally recog- 
nized that the arguments in favor of slow engines are based 
upon other points than purely mechanical ones. But one 
of the essential points in good design for high speed is 
plenty of cushion. It is therefore another one in the string 
of happy coincidences that the type of engine which natu- 
rally adopts the highest possible speed and the simplest 



238 THE THERMODYNAMICS OF HEAT-ENGINES 

possible valve-gear is forced by the latter and permitted 
by its clearance to have just what is demanded by its speed, 
viz. : a high degree of cushion. On the other hand, the 
type of engine which naturally adopts a comparatively slow 
speed and a large cylinder is permitted, by the complex 
valves and gear which it needs and can pay for, to have 
very little clearance ; consequently it may have little or no 
cushion. But cushion it does not need. 

6. Steam-jacketing. — ■ From the characteristic aims which 
have already been mentioned as outlining these two types 
of engine it is plain that, were the steam-jacket to develop 
under all conditions of operation the same average gain of 
efficiency, it would be quite likely that the large, slow-speed 
engine would find it profitable to adopt it, whereas the 
single-valve type would not. But if the conditions which 
exaggerate or minimize the effectiveness of jacketing be 
contrasted for the two types, it is seen that the four-valve 
engine possesses all of the former and the single-valve all 
of the latter. Large cylinders, slow speed, and short cut- 
off all increase the need and the effectiveness of a jacket; 
and the type which embodies these features can afford to 
pay for a jacket if it will even perceptibly aid efficiency. 
Small, compact cylinders, high speed, and moderate cut-off 
lead to moderate condensation, so that a jacket would have 
little chance of doing good ; at the same time, the engine 
embodying these features cannot afford the complication 
of a jacket, nor does it care to go after increased cylinder- 
efficiency even were it more marked than under these con- 
ditions would be the case. 

7. Condensation or Non-condensation of ExJiaust. — This 
point in design is different from the others in that its fea- 
sibility depends very largely upon a purely external and 
largely fortuitous circumstance, viz. : the existence of a 
proper supply of cheap condensing water. The amount 



THE SIMPLE STEAM-ENGINE 239 

of water needed is from one to two gallons per horse-power 
per minute. The amount of coal to be saved by its use is, 
to give a single figure which will roughly average all the 
varying conditions, not far from one pound per horse- 
power per hour. But even if it be true, in any particular 
case, that the water-supply be present and that the cost 
of condensation has been rightly interpreted, it is yet to 
be remembered, in close analogy to the preceding argu- 
ment, that the conditions which speak for the adoption of 
condensation also dictate the use of the four-valve type of 
engine. On the other hand, conditions which would lead 
naturally to the choice of a single-cylinder type of engine 
would hardly warrant the adoption of the expensive com- 
plexities incidental to working under a vacuum. The 
crossing of these natural combinations — that is, of a 
single-valve engine with a condenser, or of a four-valve 
engine with atmospheric exhaust where condensation is 
possible — may be accepted upon its face as irrational and 
wasteful. 

To summarize : There is no thermodynamic reason why 
engines of the highest speeds should not have the highest 
efficiencies. The work of Willans and Rites has proven 
this. The reasons why they usually do not are economic 
or commercial in character. 

On the other hand, there are reasons both thermodynamic 
and economic in character why the high-speed type of 
engine excels the other in sizes below 75 to 100 horse-power. 
In the writer's opinion four-valve engines smaller than this 
should become as thoroughly obsolete as should the throttle- 
governed engine of any size whatever. 



CHAPTER II 

THE COMPOUND STEAM-ENGINE 

The study of the compound engine must be approached 
from the standpoints of two very distinct sets of con- 
siderations : (i) the mechanical, and (2) the thermal. In 
the development of the steam-engine, compounding was 
attempted by Woolf at a very early date, late in the eigh- 
teenth century, in fact, but was unsuccessful owing to tar- 
diness of development of purely external considerations. 
More than half a century elapsed before McNaught 
introduced it again, and then, it appeared chiefly as a 
modification of existing engines to permit the use of 
higher steam-pressure and the development of greater 
power. The objects sought were purely mechanical. It 
was quite as a secondary and incidental result that it 
appeared that the efficiency also was increased. 

But McNaught's method of compounding applied only 
to the heavy stationary beam-engines of the first half of 
the last century, and found a very limited adoption. By 
i860, however, it had come to be a pretty well estab- 
lished fact that the larger marine engines must include 
more than one working-cylinder and crank, for purely 
mechanical reasons. Weights of individual parts were 
thereby reduced, evenness of turning-moment on the shaft 
was much enhanced, the higher rotative speeds demanded 
by the substitution of the screw for the paddle-wheel were 
facilitated, and the danger of complete failure of power 
from the failure of any one small piece in the engine was 
considerably decreased. Then, the gain in efficiency from 

240 



THE COMPOUND STEAM-ENGINE 24 1 

compounding having been recognized from McNaught's 
experience, the substitution of two cylinders of dissimilar 
dimensions using the steam in series for two cylinders of 
equal size using it in parallel was an exceedingly simple, 
easy, and inexpensive step. Nevertheless it was one of 
supreme importance for the development of the world's 
maritime trade and of the steam-engine in all services. 
Upon the increased efficiency due to the adoption of com- 
pounding in marine engines was based the tremendous 
expansion of the merchant marine in radius of steaming 
ability, in speed, and in cargo-carrying capacity. In the 
earlier days of simple engines the transatlantic voyage was 
apt to consume twenty days, and the daily consumption of 
coal was such that a vessel could scarcely carry her own 
fuel for a longer voyage ; certainly not with a proper 
margin of freight-paying cargo. But as compounding 
made the engines more efficient more power could be put 
on board, shortening the time-length of a given voyage ; the 
decreased time joined hands with the increased efficiency 
to reduce the demand upon cargo-space for fuel. 

By 1875-80 the growth of the ocean liner had attained 
such a size that propelling engines upwards of ten to 
fifteen thousand horse-power had become common. All 
of these engines were compounded, of course ; but the 
same considerations which had led to the splitting of the 
non-compound engine of 1850 into two cylinders now 
forced the adoption of three or more cylinders working 
upon three separate cranks in the design of these big com- 
pound engines. The high-pressure engine was usually 
placed in the centre, with two low-pressure cylinders for- 
ward and aft of it respectively. The cranks were usually 
set at an angle of 120 apart; the steam-pressure was 
ordinarily between 100 and 125 pounds. 

When the triple compound was introduced (1880-85) 



242 THE THERMODYNAMICS OF HEAT-ENGINES, 

as a more efficient and no more expensive way of building 
the standard three-cylinder engine, it soon led to engines 
of still larger power, because under the improved efficiency 
the limited coal-bunkers and boiler-weights would permit 
it ; and as the size of engine increased, four or more 
cylinders and cranks were adopted in triple-compounding 
just as three had been used in double-compounding. 

To-day it is an accepted fact that the marine engine of 
any appreciable size must have at least four separate 
cylinders, and often more, working three to five separate 
cranks. The steam-distribution in these cylinders may be 
upon the double, triple, or quadruple plan of compounding, 
according to the special conditions of the service in view, 
and to the designer's estimate of the thermodynamic gain 
to be expected ; but the multiplication of cylinders has 
been dictated by purely mechanical considerations and has 
lain back of and quite independent of all thermodynamic 
factors. For this reason the mechanical features of the 
question of compound vs. simple engines will be taken up 
first, although in mere outline, and the thermodynamic 
considerations left to a secondary consideration. 

Mechanical Differences. — To gain a clear idea of what is 
meant by compounding reference must be had to Fig. 42. 

Let BB' represent the given boiler-pressure, A A' the 
atmospheric line, and CC the vacuum in the condenser. 
Then, with a given supply of boiler-steam BD, the power 
available is given by the area BDC'C, supposing the curve 
DRC" to be extended to intersect CC. This power would 
ordinarily be developed in a cylinder having the total 

volume CC and cutting off at — — of the stroke. If it 

be desired to use compounding, however, a high-pressure 
cylinder may be added to the engine and an intermediate 
receiver introduced between the two cylinders. 



THE COMPOUND STEAM-ENGINE 243 

Let the pressure in the receiver be RR. This pressure 
is determined by the volume at cut-off in the low-pressure 
cylinder RR, which is, of course, capable of adjustment. 
The indicator-card of the high-pressure cylinder now 
becomes BDRR, and that of the low-pressure cylinder 
RRC'C. If it is desired to limit the effective terminal 
pressure to the profitable minimum defined on page 234, 
making both cylinders smaller will cut off the toes of the 




Fig. 42 



cards. If the volume of the high-pressure cylinder be 
reduced from RR to Gx, and that of the low-pressure 
cylinder be reduced from CC to Fy, the toes of the cards 
will be cut off at xx l and yy' respectively. 

In triple compounding there would be two such dividing 
lines as RR, representing the pressures in the first and 
second receivers respectively ; in quadruple compounding 
there would be three such horizontal lines. 

The first and most obvious conclusion from this is that 
the total area of available work is no greater after com- 



244 THE THERMODYNAMICS OF HEAT-ENGINES 

pounding than it was before. Indeed, it is somewhat 
smaller by the loss of pressure inevitable in passing the 
steam from one cylinder to another. From this broad 
fact is deduced the rule that — 

In a given compoimd engine, under a given boiler-presstire 
and a given ratio of expansion, the poiver developed is inde- 
pendent of the number of cylinders through which the steam 
is worked before reaching the low-pressure cylinder ; it is 
the same as if the same boiler-steam performed the same 
number of expansions in tlie low-pressure cylinder alone. 

The simplest form of compound engine is the tandem, 
where both pistons work upon the same piston-rod and 
crank. With the exception of the compound beam-engine 
of McNaught, which never found a wide acceptance, 
although some very large engines have been built on his 
plan, it was also the first. It is undoubtedly the cheapest 
to build, and so finds wide acceptance yet, for stationary 
engines of moderate power. Its latest field of adoption is 
in locomotive work. Here it contests for the laurels with 
the Vauclain arrangement of a pair of parallel cylinders, 
high-pressure and low-pressure, on each side of the loco- 
motive, each pair working on a single cross-head or with 
the several European types of four-cylinder compounds. 
The two-cylinder compound locomotive, with a high-press- 
ure cylinder upon one side and a low-pressure cylinder on 
the other, does not promise to be a permanent type. 

The purely mechanical advantages of compounding have 
been stated as consisting of lighter weights all around, a 
more even turning-moment, better equalized strains upon 
all working-parts, and greater ease in properly distributing 
the steam. It would seem, at first sight, that in the tandem 
compound none of these advantages could appear; for 
since a single crank and set of connections is relied upon 



THE COMPOUND STEAM-ENGINE 245 

to develop the same power in either case, the difference 
would appear to lie wholly within the cylinders, and that 
their complexity and weight must be doubled. Upon in- 
vestigation, however, such does not appear to be the case. 

Referring to Fig. 42, let a simple-cylinder engine be 
compared with a tandem-compound, supposing that in each 
case the boiler-pressure be 145 pounds by gage, or 160 
pounds absolute, and that the final volume Fy, in order to 
develop the desired power, must be 10 cubic feet. This 
may be considered as supplied by a cylinder 27" x 30", 
which would constitute the only cylinder of the simple 
engine and the low-pressure cylinder of the compound. 
To the latter must be added the high-pressure cylinder 
upon the same axis with the low-pressure cylinder, with 
their pistons attached to a common piston-rod. If the 
proper ratio between the two be assumed to be four to one, 
this would call for a high-pressure cylinder 13 J" x 30". 

In the case of the simple engine the maximum pressure 
upon the piston, which would occur just at the beginning of 
the stroke, would be (neglecting back-pressure) 160 pounds 
over an area of 572 square inches, or a total of 91,520 
pounds. The cylinder, piston, piston-rod, connecting-rod, 
crank, shaft, and engine-bed must all be designed to with- 
stand this stress. 

In the case of the compound (if the simple empirical 
equation of pv = a constant be assumed as a basis for esti- 
mates) the receiver-pressure should be ^ of 160 = 40 pounds 
absolute. Then the high-pressure piston would be subjected 
to a maximum stress of 120 pounds over an area of 143 
square inches, or a total of 17,160 pounds. The low-press- 
ure piston would be subject to a maximum pressure of 40 
pounds over an area of 572 square inches, or a total of 
22,880 pounds. 

It is immediately obvious that, in so far as cylinders and 



246 THE THERMODYNAMICS OF HEAT-ENGINES 

pistons alone are concerned, they need to be built to with- 
stand a stress only about one-quarter as large as before. 
Not only will they be much cheaper to build and handle, 
but all their appurtenances in the way of valves and gear 
will be much lighter, cheaper, and more reliable, and engine- 
friction will be much reduced. Even when attention is 
turned to cross-head, crank, and connections a great gain is 
visible. The total maximum stress upon them in the case 
of the compound is 40,040 pounds as compared with 91,520 
pounds for the simple engine, representing a gain of over 
60%. 

The explanation of the great difference, for the same 
power developed on a single crank, lies in the fact that 
whereas in the simple engine the maximum stress would 
continue only throughout about one-sixteenth of the stroke 
(if a terminal pressure of 10 pounds be assumed), in the 
compound the cut-off would not take place until one-quarter 
of the stroke had elapsed. The pressure would be more 
moderate in the case of the compound, but would be more 
continuously sustained. 

If attention be turned from the tandem-compound to the 
cross-compound, where each cylinder has its own crank, or 
to the marine-type of compound, where the work developed 
in a single one of the stages is often distributed between 
two cylinders and cranks, the degree to which this gain in 
lightness and strength may be exaggerated can be well 
imagined. In the last-named type, too, an additional argu- 
ment for multiplication of stages of compounding, as well 
as of cylinders and cranks, lies in the fact that these engines 
can carry no fly-wheel and possess light foundations, and 
therefore require a very even turning-moment on the shaft. 
A short cut-off under high pressure upon a single large 
piston would start off the stroke with a disastrously violent 
jump. At the other end of the stroke, too, since the re- 



THE COMPOUND STEAM-ENGINE 247 

versing valve-gear under which all marine engines must 
work introduces a heavy cushion as an inevitable accompa- 
niment to an early cut-off, the piston would be arrested in 
its motion by an excessive negative pressure. The engine 
would either fail to pass dead-centre altogether, or would 
do so reluctantly and with noise or vibration. Compound- 
ing remedies all of these troubles. As even so moderate a 
cut-off as one-quarter stroke, however, would develop a 
troublesome cushion, the designer of a marine engine to 
work under a pressure as high as was assumed in the above 
illustration would choose between the alternatives of a less 
complete expansion on the one hand or of the extra cost 
and complexity of triple-compounding on the other. Thus, 
in the illustration, a terminal pressure of 10 pounds absolute 
was assumed in the low-pressure cylinder and complete ex- 
pansion to receiver-pressure in the high-pressure cylinder. 
This would make the indicator-card of the former follow 
the diagram RRyy'C, Fig. 42, and the latter the diagram 
BDRR. If, however, the stroke of each piston were 
shortened from 30 to 24 inches, the terminal pressure in the 
high-pressure cylinder would be raised to 

——.BO = — x 160 = 50 pounds, 
Gx 24 

and in the low-pressure cylinder to 

——' RO =2i2 x 40= 12.5 pounds 
Fy 24 

absolute, or 10 pounds net; which is much better engi- 
neering practice. The cut-off would have been lengthened 
from 25 to 31% of the stroke and the excessive cushion 
have been moderated thereby. 

Let it be supposed again, to take up the second alterna- 
tive in comparison, that this same expansion were to be 



248 THE THERMODYNAMICS OF HEAT-ENGINES 

carried out in a four-crank, four-cylinder, triple-compound 
engine, having one high-pressure, one intermediate, and two 
low-pressure cylinders. Since the last stage of the expan- 
sion is to be divided between two cranks, a little extra pro- 
portion of expansion and of power may be thrown into the 
low-pressure stage. This could be done, for instance, by 
giving the engine two low-pressure cylinders of 19 inches 
in diameter, one intermediate cylinder 1 5 inches in diameter, 
and one high-pressure cylinder 10 inches in diameter; the 
stroke of all Of them may be considered as still 30 inches. 
It is not to be considered by the student that this necessa- 
rily represents the best engineering practice, but it enables 
a simple comparison to be drawn between engines of widely 
different type under the same steam-pressure and expan- 
sion, in order to bring out most plainly one or two particular 
points; viz., comparative maximum stresses and compara- 
tive cut-offs. The dimensions chosen give a ratio between 
high-pressure and intermediate of 2.31, between intermedi- 
ate and low-pressure the same, and 3 as the ratio of ex- 
pansion within the low-pressure cylinder. This places the 
first-receiver pressure at 160 -=- 2.31 = 69.3 pounds, and the 
second at 69.3 -=-2.31 = 30 pounds absolute. The maximum 
pressure falling upon the high-pressure piston and crank is 
(160— 69.3)x 78.5 = 7120 pounds. That upon the inter- 
mediate piston and crank is (69.3 — 30) x 177 = 6956 pounds. 
That upon each low-pressure piston and crank is 30 x 283 
= 8490 pounds. If these figures be compared with the 
maximum stress on a single crank of 17,280 pounds for the 
two-crank cross-compound, 40,320 pounds for the tandem- 
compound, and 92,160 pounds for the simple engine, the 
gain is obvious. The cut-off in each of the two smaller 
cylinders of the four-cylinder triple-compound would be 
43-3% an d in the largest cylinders 33.3%. 

Finally, were the strokes again reduced to 24 inches i: 



;r 

: 



THE COMPOUND STEAM-ENGINE , . 249 

order to give a profitable terminal pressure and to minimize 
cushion, the cut-offs would lengthen from 43% to 54% and 
from 33% to 42% respectively. 

Although intended for illustration only, these propor- 
tions, if modified so as to give a shorter stroke and larger 
diameter of cylinder and worked under a higher steam- 
pressure than 145 pounds by gage, would represent a close 
approach to good marine-engine practice. 

Thermal Considerations 

It will be remembered that in the discussion of cylinder- 
condensation and its kindred phenomena it was noted that 
both theoretic efficiency and condensation increase as the 
temperature-range within the cylinder increases. The 
former increases almost proportionately to the tempera- 
ture-range ; the latter, insignificant at first and increasing 
very slowly with the first growth of temperature-range, 
takes on an ever-increasing augmentation as the additions 
to the temperature-range accumulate, in somewhat geo- 
metric ratio, until finally cylinder-condensation becomes 
the all-important phenomenon, overbalancing and eclipsing 
all other considerations. In other words, as the theoretic 
efficiency increases, the cylinder-efficiency decreases, at 
accelerating rate, until the resultant thermodynamic effi- 
ciency, at first ascendant, becomes stationary and finally 
declines. A temperature-range a little beyond that at 
which the latter is a maximum dictates the limits for the 
proper use of simple expansion in a single cylinder. 

If any higher efficiency than that just described is 
desired, compounding must be resorted to. But it is imme- 
diately evident that compounding under a total tempera- 
ture-range and pressure-range (between boiler and exhaust) 
which gives a maximum efficiency in a single cylinder must 



250 THE THERMODYNAMICS OF HEAT-ENGINES 

result in a loss rather than a gain ; for compounding is 
expensive, not only in cost of construction but in incidental 
losses of heat as well. The steam cannot be transferred 
from one cylinder to another without distinct loss, in fric- 
tion and in radiation. Even at a moderate advance of 
temperature-range over the point just mentioned these 
losses must overbalance the gains due to 

(i) The increased theoretic efficiency, and 
(2) The increased cylinder efficiency 

due to dividing a greater total temperature-range into two 
portions, each of which is smaller than that of the simple 
engine. The first conclusion to be observed, therefore, is 
that compounding cannot be profitable except under a total 
temperature-range decidedly greater than that specified as 
profitable for a single cylinder on page 228. This is the 
same as saying that compounding cannot possibly be 
profitable under less than 125 pounds of boiler-pressure 
when non-condensing, nor under less than 90 pounds con- 
densing. 

That is the lower limit. The upper one is probably 
about 175 pounds when condensing, or about 250 pounds 
when non-condensing. • For it is not until these figures are 
surpassed that the temperature-range in a single cylinder 
becomes great enough to warrant the introduction of a 
third stage of expansion. Where the latter policy has 
been adopted under pressures similar to the above it has 
either been for mechanical reasons, as in the marine- 
engine, or in error of judgment, as has come to be 
generally admitted in connection with triple-expansion 
stationary engines. 

It is therefore obvious that in the design of a compound 
engine, in the choice of boiler-pressure under which it 
should work and of cylinder-ratio to be adopted, there is 



THE COMPOUND STEAM-ENGINE 25 I 

the widest possible field for variation of individual judg- 
ment. There will be found a correspondingly wide varia- 
tion in existing practice. In general the problem of the 
determination of exact cylinder-dimensions would be at- 
tacked somewhat in this way. 

Problem : To design a compound engine of 100 horse- 
power normal load, to operate under a steam-pressure of 
not over 150 pounds and to exhaust into a vacuum. It is 
assumed that the general type of engine, the speed, the 
valve-gear, and all similar points outside of thermodynamic 
ones are already determined ; that the engine is to be of 
the four-valve, receiver type, and to run at 100 revolutions 
per minute. What should be the cylinder-dimensions ? 

According to the rule stated on page 244, the size of 
low-pressure cylinder must first be determined as if there 
were to be no high-pressure cylinder present, the full 
degree of expansion taking place supposedly in the single 
cylinder. Since condensation and reevaporation so modify 
the true adiabatic process that its pressure-volume repre- 
sentation closely agrees with the curve 

PV = a constant, 

it is common practice to adopt this as the basis for the 
projection of indicator-cards in such work as this ; partly 
because the entire problem can never be exact, in either 
statement or solution, and partly because the simplicity of 
the curve lends itself to the rapid investigation of the sev- 
eral supposed cases which usually need comparison. 

It would first be decided that if 150 pounds by gage 
were to be the normal boiler-pressure, 145 pounds by gage 
or 160 pounds absolute are all that could be expected as 
the initial pressure within the cylinder, considering varia- 
tions in boiler-pressure and pipe-friction. Next, the most 
advantageous terminal pressure would be chosen ; say at 



252 THE THERMODYNAMICS OF HEAT-ENGINES 

8 pounds absolute. This establishes the total ratio of 
expansion at lf-2- = 20. Let it be assumed that the vacuum 
to be expected in the cylinder is 25 inches, or 12J pounds 
below atmospheric pressure, or 2\ pounds absolute. 

Referring to Fig. 42, the expansion to the final terminal 
pressure at y, when exhaust into the condenser takes place, 
would develop the work-diagram CBDyy'. Of this, the 
area BDyF is given by the formula (deduced for this curve 



Fig. 42 

on page 122): Work = P\V X log r. The final volume at- 
tained by the steam, at the end of the stroke in the largest 
cylinder, will be represented by Fy. Let this be repre- 
sented by V h in cubic feet. Then, since Pi — 8 x 144 = 
1 1 52 pounds per square foot and r= 20, the work repre- 
sented by the area BDyF must be 11 52 V l log e 20=3451 V x 
foot-pounds. The work represented by the area CFyy' is 
(8 — 2^) 144 Vi = 828 Vi foot-pounds. The total is 

4279 V x foot-pounds. 



THE COMPOUND STEAM-ENGINE 253 

Since the engine is to be of 100 horse-power, the work 
done per minute must be 3,300,000 foot-pounds. Placing 
this equal to the expression just found and solving for V h 
gives the value of the latter as 771 cubic feet. This must 
be the volume displaced by the piston each minute. If 
the engine is to make 100 revolutions, or 200 strokes, per 
minute, the displacement per stroke must be 3.85 cubic feet. 

This volume may be attained by any proportion of diam- 
eter to stroke which suits the mechanical sense of the de- 
signer. The chief guide in the choice is the speed : the 
slower the speed, the longer the stroke ; the higher the 
speed, the shorter the stroke. A good proportion for this 
case would seem to be given by a cylinder if x 30", 
which gives slightly more than the desired displacement. 1 

Receiver-pressure and Cylinder-ratio. — Next arises the 
question as to the size of the high-pressure cylinder. This 
is determined by the choice of receiver-pressure and of 
terminal pressure in the high-pressure cylinder. 

The simplest solution of the case is to consider that the 
total ratio of expansion is to be divided equally between the 
two cylinders ; that is, that the square root of the total ratio 
is to be the ratio in each cylinder. In Fig. 42 RR is the 
receiver-pressure and xx' the exhaust-line at the termination 
of the high-pressure stroke. By the above argument the 
pressure at x would be 

OG = OFyjy^-= VOF x OB = V8 x 160 = 35.8 pounds. 
Similarly, 

OR = OCyjy-^ =VOCx OB= V2. 25 x 160 = 19 pounds, 

1 It is to be remembered that no consideration is being given here to the 
effects of clearance and cushion upon the ratio of expansion. Their inclusion 
would probably bring the figures up to about iS" X 30". 



254 THE THERMODYNAMICS OF HEAT-ENGINES 

or the receiver-pressure would be 4.5 pounds by gage. 
The volume at x is 

z~* r* OF 7-877' 

CrX = by • = by • = by 



OG 35.8 4.475 

It is stated in this way because 4.475 is therefore the de- 
sired ratio between the volumes of the two cylinders ; but 
as the strokes of the two are nearly always the same, the 
ratio of diameters may be taken as the square root of the 
ratio of volumes. This gives, for the diameter of the high- 
pressure cylinder, 



17 -T- V4.475 = 8.036, or, say, 8 inches. 

This gives, as the proper cut-off in the low-pressure 
cylinder, RR -s- Fy = OF-r- OR = 22%, and in the high- 
pressure cylinder BD -s- Gx = OG -s- OB = 22%. 

Another common rule for determining the receiver-press- 
ure is to divide the work done equally between the two 
cylinders. 

The total work was found to be 4279 V 1 foot-pounds per 
minute. Half of this is 2140^ foot-pounds. Deducting 
the area CFyy' , the remainder of the low-pressure area, 
FRRy, is 1312^ foot-pounds per minute. Since this is 
equal to P 1 V 1 log r, and since P 1 = 8 x 144, r is easily 
found to be equal to 3.125. OR = 8 x 3.125 = 25 pounds 
absolute, as compared with 19 pounds by the former 
method. 

The area remaining for the high-pressure cylinder to 
develop would be RBDR. Some triangular portion of 
this, as xRx', would properly be cut-off in order to give 
that cylinder an appreciable and profitable terminal press- 
ure, and would be lost ; but as it is small compared to the 
whole, and as the rating of 100 horse-power taken as a 
basis for all of the calculations is a purely nominal one, 



THE COMPOUND STEAM-ENGINE 255 

such incidental minor losses are usually neglected for the 
time, and are allowed for in lump by increasing the calcu- 
lated cylinder-dimensions by the fractions necessary to 
bring them up to some convenient integral figure. Thus, 
if 20 pounds be chosen as a proper high-pressure terminal, 
the total pressure at x would become 45 pounds, and the 
cylinder-ratio would be Fy-h-Gx— OGs-OF=4$^-8= 5.625, 
which calls for a high-pressure cylinder about J^q inches 
in diameter. 

But any and all of these methods, of which there are 
several, are rule-of-thumb substitutes for the only rational 
thermodynamic one, viz. : the equal division of tempera- 
ture between the cylinders. In the present case the total 
temperature-range, from 160 to 2\ pounds absolute, is 
232. 8°. Half of this is 116.4 , which, added to the back- 
pressure temperature in the low-pressure cylinder, gives 
247 F. as the proper receiver-temperature, or 28.3 as its 
proper absolute pressure. A high-pressure net terminal 
of 20 pounds again would make the gross high-pressure 

terminal at.r, 48.3. The cylinder-ratio would be ^—^ = 6.04. 

The diameter of the high-pressure cylinder would be 
17 -r- V'6.04 = 6\^" ; say 7 inches. 

It is obvious that all of these methods lead to very 
similar conclusions. The second one is very commonly 
resorted to, originally because it was very easily carried 
out by means of the indicator. It rests upon good foun- 
dation in those cases where mechanical considerations out- 
weigh thermodynamic ones ; but the false idea has grown 
out of this practice, and been widely taught to operating 
engineers, that the equal division of load between the 
cylinders is thermodynamically proper. It is usually me- 
chanically proper, though sometimes not. It is only a co- 
incidence if the thermodynamic results happen to be good. 



256 THE THERMODYNAMICS OF HEAT-ENGINES 

Clearance and Cushion. — The influence of clearance and 
cushion upon capacity, or upon ratio of expansion, has 
been openly excluded from the above calculation. The 
method of allowing for it was explained in Chapter I, page 
196. Suppose that in the low-pressure cylinder of the above 
engine it be known that the actual clearance (which would 
be estimated from a knowledge of the type of valve-gear) 
would be about 2%. The calculations on page 196 showed 
that such a clearance necessitated the increase in volume 
of the cylinder, in order to get the same results from the 
same, quantity of steam, from 2.62 to 2.77 cubic feet, or by 
about 6%. Such an allowance for clearance as this would 
call for the increase of the 17-inch cylinder to 17I inches 
in diameter. The high-pressure cylinder should of course 
be similarly increased in diameter, to correspond. 

The preceding problems cover the bulk of what may be 
stated with mathematical accuracy regarding the choice of 
cylinder-ratio in compounding. In the remaining portion 
of the field, which is open to variation of opinion, there is 
the widest divergence between the judgments of men of 
the highest standing. The Westinghouse compound sta- 
tionary engine and the Vauclain compound locomotive, 
both of which have been exceptionally successful designs, 
use a ratio of about 2.7 to 1 under as high as 200 pounds 
boiler-pressure, but with no vacuum. Mr. George I. Rock- 
wood advocates 7 to 1 as a proper ratio, under even lower 
boiler-pressures when exhausting into a vacuum, and his 
engines have given good results. Between these two ex- 
tremes might probably be assorted all the different opin- 
ions regarding compound-cylinder ratios in the country. It 
would seem unquestionable, however, in spite of this wide 
divergence of opinion, that a broad survey of the progress 
of experience in engine-design reveals the following facts : 



THE COMPOUND STEAM-ENGINE 257 

(1) That the steam-pressure which can be efficiently 
expanded in two stages is higher than was formerly sup- 
posed. The need for the intermediate cylinder in the 
triple compound is based upon purely mechanical, not 
thermodynamic, principles. The maximum cylinder-effi- 
ciency is attained in each cylinder of a compound en- 
gine under approximately the same conditions as in the 
cylinder of a simple engine, viz. : under from four to 
six expansions. These three statements are practically 
synonymous. 

(2) That, as a result, cylinder-ratios are steadily upon 
the increase. 

The favorite ratio in England was, for many years, for 
double compounds four to one and for triple compounds 
six to one, as between low- and high-pressure cylinders. 
This applied normally to large condensing compounds. 
Compounding was adopted more reluctantly in America 
and at first copied English practice in this respect. By 
1895, however, the plan of carrying a standard line of 
compound designs and patterns for all services was well 
established. As the majority of these engines were sold 
to operate either under steam of abnormally low pressure 
from existing boilers, or under atmospheric exhaust, the 
ratio of four to one was found to be too large. General 
practice soon settled down to the region between 2.75 and 
3.25 to one. Within the past few years, however, the 
trend is back toward the higher ratios, and values above 
four to one are now quite common in new engines from 
the best works. 

To the marine engine, where mechanical considerations 
outweigh thermodynamic ones, these remarks, of course, 
do not apply. Cylinder-ratios are much lower there, in 
order to avoid too early cut-offs and too heavy cushion for 
engines without fly-wheels. 



258 THE THERMODYNAMICS OF HEAT-ENGINES 

The Receiver-type and the Woolf-type of 
Compounds 

It is almost universal practice, in compounding, to ex- 
haust from the high-pressure cylinder into a chamber of 
considerable volume, called a receiver, and to draw from it 
the supply for the low-pressure cylinder as from a boiler. 
The advantages resulting from this plan are several. They 
may' be lumped under two heads, as follows : — 

(1) Mechanical. — The timing of the motions of the two 
pistons is quite independent one of the other. The cranks 
driven by the two cylinders may be set at any desired 
angle. 

(2) Thermodynamic. — The pressure in the receiver re- 
mains nearly constant ; hence the temperature-range in the 
high-pressure cylinder is only that from boiler to receiver 
temperature, and the temperature-range in the low-pressure 
cylinder is only that from receiver to exhaust temperature, 
Were this not so, were there a variation in the receiver- 
pressure, the high-pressure temperature-range would extend 
from boiler to the lowest receiver-temperature, while the 
low-pressure range would be from the highest receiver- 
temperature to exhaust. The two ranges would lap, their 
sum would exceed the total range from boiler to exhaust, 
and the resultant condensation would be greater than it 
otherwise might be. 

This is the condition of affairs in the Woolf-type of 
compound. It has no receiver. The two cylinders are 
separated only by the exhaust and admission valves. This 
necessitates — 

(1) That the pistons must move simultaneously. (They 
may move in opposite directions, however.) 

(2) The intake of the larger cylinder can consist only of 
the exhaust from the smaller ; the difference in size be- 



THE COMPOUND STEAM-ENGINE 



259 



tween the two pistons therefore insures that the " receiver- 
pressure " must steadily decrease, owing to expansion, 
during the exhaust-stroke of the high-pressure and the 
admission-stroke of the low-pressure cylinder. These phe- 
nomena are visible in Fig. 43, in which the dotted lines 
show the effect of shortening the cut-off from that shown 
by the full lines. 1 

The Woolf engine was one of the earliest compounds to 



BOILER- PRESSURE. 



HIGH-PRESSURE CYLINDER; 
CLEARANCE, 33$. 



LOW-PRESSURE CYLINDER; 
CLEARANCE, if' 



ATMOSPHERIC PRESSURE. 



Fig. 43 

appear, late in the eighteenth century. Unable to suc- 
cessfully compete with Watt's monopoly or to establish its 
superiority under the low steam-pressures then prevailing, 
it failed and disappeared. Within the last twenty years, 
however, its marked superiority under certain conditions 
has been appreciated and it has reappeared. 

For the reasons just given, the receiver-type of com- 

1 See Trans. Am. Soc. M. E., Vol. XIII, " Steam-distribution in a Single- 
acting Engine," F. M. Rites. 



26o THE THERMODYNAMICS OF HEAT-ENGINES 

pound is superior, both in mechanical adaptability to 
diverse arrangements of piston-motion and in thermal 
efficiency, to the Woolf-type, under nearly all conditions 
of service. It is therefore only necessary, in order to 
properly define the worth of the latter, to consider the 
conditions under which the receiver-type is not a success. 
These may be very easily and narrowly defined as a 
light or variable load combined with atmospheric exhaust. 
Under these conditions the efficiency of the receiver-type 
diminishes so very rapidly as to easily fall below that of 
the simple engine, and sometimes below the poorest which 
could ever be expected from a good simple engine. 

The reason for this is that under light loads the governor 
admits to the engine so small a volume of boiler-steam that 
the terminal pressure in either or both cylinders falls below 
the exhaust-pressure. The result is a loop at the lower 
end of the card, the pressure in the cylinder rising when 
the exhaust-valve is opened, instead of falling, as it should. 
While a loop at the upper corner of the indicator-card 
signifies little loss of efficiency, if any, a loop at the lower 
end of the card is trebly bad. 

(i) It much exaggerates the evils already described as 
arising from complete expansion to exhaust-pressure, on a 
sharp-toed diagram. 

(2) It absorbs power excessively which is not returned 
to the engine in the slightest appreciable degree. 

(3) It exaggerates cylinder-condensation by sucking in 
through the exhaust-port moisture which, much of it, must 
reevaporate within the cylinder at the expense of heat made 
good from the next initial condensation. 

Trouble (2) could theoretically be cured by equipping 
the cylinder with check-valves opening back from exhaust 
into the cylinder whenever the pressure in the latter fell 
too low ; but they would be noisy, frail, and troublesome,. 



THE COMPOUND STEAM-ENGINE 26 1 

and would have no effect on the other evils. They have 
never been widely applied to mill-engines, though common 
in locomotives. 

The effect of these losses is pernicious in the extreme. 
In one case it was reported, by a high authority, that a 
good specimen of Corliss compound engine, new and in 
good condition, which would ordinarily be expected to 
develop a horse-power upon about 14 to 15 pounds of steam 
per horse-power-hour when properly loaded and supplied 
with a vacuum, actually consumed, when lightly loaded 
and exhausting into the atmosphere, 72 pounds of steam 
per horse-power-hour. If the low-pressure cylinder had 
been disconnected, the other cylinder alone would have 
developed the power, under these conditions, at a rate not 
much exceeding one-third of this figure. It did develop 
about 180 horse-power at just about that rate ; but the low- 
pressure cylinder, acting as a brake by reason of its having 
to drive its exhaust out against an atmospheric pressure 
which was higher than its own terminal pressure, absorbed 
over 100 horse-power, so that the net rate was that 
stated. The high-pressure cylinder alone would not, of 
course, have developed the 70 or 80 net horse-power as 
efficiently as it did the 1 80 horse-power ; but the rate, even 
so, could hardly have risen above 30 pounds per horse- 
power-hour. 

The superiority of the Woolf-type under these same con- 
ditions is due to the following circumstances : — 

There being no volume between the two cylinders, any 
cut-off in the low-pressure cylinder must also act as the 
closing of the exhaust of the high-pressure cylinder, or as 
the point of compression in that cylinder. The fact that 
an ordinary cylinder cannot be given more than a certain 
degree of cushion (without exceeding boiler-pressure before 
admission takes place) has led to the oft-stated but false 



262 THE THERMODYNAMICS OF HEAT-ENGINES 

idea that the low-pressure cylinder of a Woolf compound 
could have no cut-off. But any cylinder may be given as 
early a cushion as is desired, provided there be sufficient 
clearance to absorb the compression before too high a ter- 
minal pressure is reached. Therefore all modern designs 
of Woolf-type compounds are provided with an unusually 
large clearance in the high-pressure cylinder. This per- 
mits an early cut-off in the low-pressure cylinder and an 
early, slowly rising cushion in the high-pressure cylinder. 

If this cushion were sufficient to engender boiler-pressure 
in the clearance space before admission took place, and if 
there were no thermal losses to cylinder walls, this large 
clearance would involve no loss of efficiency (see page 206; 
236). In practice neither condition is supplied. Losses 
are involved from both sources. The large clearance must 
be partially refilled with boiler steam before each stroke 
commences ; owing to thermal interchanges between the 
steam and walls the cushion-steam does not return as much 
work during expansion as it absorbed during compression. 
Therefore the Woolf-type never attains the maximum effi- 
ciency usual in receiver-compounds when run under proper 
conditions. Nevertheless, the sum of these two losses in 
the Woolf-type is not nearly so great as is that incurred by 
the over-expansion of the receiver-type under light loads 
and no vacuum. The latter loss can be entirely avoided in 
the Woolf-type by making the high-pressure clearance suf- 
ficiently large, so that under the lightest possible load, — 
zero load, say, for an extreme illustration, — there will 
always be steam enough in the engine to expand through 
both cylinders and still be above atmospheric pressure at 
the end of the low-pressure stroke. Since the cushions in 
both cylinders are supposedly so designed that the ordinary 
back-pressure, while the exhaust is open to the atmosphere, 
is compressed back to boiler-pressure in the high-pressure 



THE COMPOUND STEAM-ENGINE 263 

clearance, there would be, theoretically, in the extreme 
case of no load just supposed, no intake of steam from 
the boiler at all. In practice this nice adjustment of 
valve-motions is found impracticable, and some steam is 
always consumed in merely refilling the high-pressure 
clearance at each stroke ; but the approach to the theory 
is quite close. 

The origination of this plan of operation the world owes 
to Mr. F. M. Rites, who incorporated it into the design 
of the Westinghouse compound engine, which is a pure 
Woolf-type engine. It is also embodied in modified form 
in the four-cylinder Baldwin compound locomotives of Mr. 
Vauclain's design. Since this beginning of its revival 
other designs of Woolf-type engines have been placed 
upon the market. 

Methods of Steam-distribution 

Compound engines may be rigged with any of the 
several types of valve-gear used on simple engines : The 
single balanced-slide or piston-valve, driven either by a 
fixed eccentric (with throttle governor), by a shaft-gov- 
ernor, or by link-motion ; the four valves of the Corliss 
engine or their equivalent, closed either by drop cut-off 
gear or positively ; or the double slide or piston- valves of 
the Myers-type. But the differences between these gears 
are much the same in compound as in simple engines. 
Their discussion belongs to a study of valve-gears, not 
that of the thermodynamics of compounding. 

There is one question calling for decision in such work, 
however, which has direct thermodynamic influence upon 
the efficiency. That is : Shall the cut-off in the low-press- 
ure cylinder be varied by the governor as the load changes, 
or shall it remain fixed ? 

There is much needless discussion of the question, chiefly 



264 



THE THERMODYNAMICS OF HEAT-ENGINES 



because of doubt as to what result is desired, rather than 
as to what will ensue from a given course of action. For 
it is plain that if the low-pressure cut-off remains fixed 
while the steam admitted to the high-pressure cylinder is 
varied with the varying load, the receiver-pressure must 
also vary with the load. When the latter is heavy the 
receiver-pressure will rise ; when the load is light, it will 
fall. If, on the other hand, the cut-offs of the two cylin- 



A 

c 

E 


b' B 


b 






\ \ 

\ \ 

— r~ 
\ 


\ 

\ 
\ 

\ 
\ 

1 

\ 
\ 

D 


a 

\ 

s 
c 


— 

■ — . 




\ 


"d ; 


O 


G 


K 






H 






O 








V 



Fig. 44 



ders were varied together and equally by the governor as 
the load altered, the receiver-pressure would remain prac- 
tically constant. 

I. Low-pressure Cut-off Fixed. This situation is shown 
by Fig. 44, wherein, under normal load, the governor ad- 
mits the volume AB of boiler-steam to the engine. ABCDE 
is the high-pressure indicator-card and EFGHK is the low- 
pressure card. 

When the load increases, the governor admits the volume 



THE COMPOUND STEAM-ENGINE 



265 



of boiler-steam Ab. Its curve of expansion is bcfg. The 
high-pressure card becomes Abcde and the low-pressure 
efgHK. 

Under lighter loads the expansion-curve becomes b'c'f'g' 
and the receiver-pressure falls to e ! f. The high-pressure 
card becomes Ab'c'd'e' and the low-pressure card becomes 
e'f'g'HK. 

II. Low-pressure Cut-off always Equal to High-pressure 
Ciit-off. This situation is shown in Fig. 45. The normal-load 
p 




Fig. 45 



diagrams are ABCDE for the high-pressure and EFGHK 
for the low-pressure, as before. 

An increase in load moves the expansion-curve out to 
bcfg, as before ; but because the low-pressure cut-off has 
lengthened to f the receiver-pressure does not rise. The 
high-pressure card becomes AbcDE. The low-pressure 
card becomes EfgHK. 

Under lighter loads the expansion-curve becomes b'f'c'g' ; 
but again, because the low-pressure cut-off has shortened 



266 THE THERMODYNAMICS OF HEAT-ENGINES 

from Ftof, the receiver-pressure remains the same. The 
high-pressure card becomes Ab'f'c'DE. The low-pressure 
card becomes Ef'c'g' HK. 

Comparative Efficiency. — The chief differences between 
the two plans are obvious. The primary advantage of the 
first is that under all loads the small triangular area CDF, 
which represents receiver-loss, remains practically constant. 
It is entirely under control and is fixed in the original de- 
sign of the engine and the set of the low-pressure cut-off. 
If the volume in the low-pressure cylinder at the instant of 
cut-off were equal to the cylindrus of the high-pressure 
cylinder, this loss-area would be zero under all loads ; by 
making the high-pressure cylinder a little smaller than this, 
some cost and friction can be saved, at the expense of no 
appreciable loss of power or efficiency, and the receiver- 
drop CD can be made whatever is desired. 

In the second plan, on the other hand, the receiver-loss 
varies widely. Under heavy loads it grows to cDf. As 
the load falls below normal it decreases first to zero and 
finally becomes negative, an extreme which is far more 
harmful to efficiency than its opposite. The terminal loss 
at GH or gH is alike in either plan. So that the thermo- 
dynamic advantages lie all with the first plan. 

In respect to mechanical features, however, it is quite 
otherwise. Under the second plan the work is divided 
between the two cylinders quite evenly under all loads. 
Under the first, on the other hand, heavy loads are de- 
veloped almost entirely in the low-pressure cylinder and 
light loads almost entirely in the high-pressure. This is 
mechanically bad. It could be allowed for, however, by 
proper design and construction. But this distribution of 
power has a secondary and very detrimental effect upon 
another important feature of engine-operation, viz. : — 

Regulation. — Under the first plan the receiver-pressure 



THE COMPOUND STEAM-ENGINE 267 

is varied up or down by the constant volume-pull of the low- 
pressure cylinder upon a varying supply of steam to the 
receiver from the high-pressure cylinder. The receiver- 
pressure will therefore be constant after any alteration of 
load, only after these two factors have had time to attain a 
condition of equilibrium. In other words, when the load 
decreases, the low-pressure cylinder finds itself supplied 
with steam of a much higher pressure than that of equi- 
librium. It therefore develops more power than it would 
after the steady conditions which are normal to this load 
have been attained. But, the load having fallen off, this is 
just the time when less power, not more, is wanted from the 
low-pressure cylinder. The engine therefore jumps above 
speed until the excessively short high-pressure cut-off in- 
duced thereby, coupled with the heavy draft of the low- 
pressure cylinder on the receiver, has brought the pressure 
of the latter to the proper point. 

But the process cannot stop there, for equilibrium is not 
yet attained. The engine-speed is above normal ; the high- 
pressure cut-off is shorter than that suited to the load. The 
receiver-pressure therefore continues to fall below that of 
equilibrium until arrested and brought back by a reversal of 
the entire above process, which will be instituted only when 
the engine-speed has again fallen below normal, a thing 
which must finally result from the too short cut-off in the 
high-pressure cylinder. 

This process of wavering above and below normal speed 
at every change of load is called "hunting." There are a 
long list of causes of hunting which lie in the mechanics of 
the governor. But the effect of all such causes will be 
much exaggerated by compounding with fixed low-pressure 
cut-off. Indeed, so extreme is this exaggeration in some 
cases that there is no position of equilibrium possibly to be 
found ; the engine surges above and below normal speed 



268 THE THERMODYNAMICS OF HEAT-ENGINES 

so violently that operation under loads which vary appreci- 
ably is quite impossible. 

Extreme conditions may dictate the adoption of either 
plan unreservedly. That is, extreme demand for perfect 
regulation may enforce the full variation of low-pressure 
cut-off with that of the high-pressure cylinder, regardless 
of the effect upon the efficiency ; or the need for the most 
refined efficiency may call for the abandonment of all hope 
of close regulation. Usually, however, a compromise is 
wise. The attachment of the low-pressure cut-off to the 
governor will insure moderately close regulation, but by 
diminishing the extent and rapidity with which it is varied 
by the governor a good grade of efficiency is maintained. 

Reheaters. — It will be remembered, from the theory of 
thermodynamics, that the adiabatic expansion of saturated 
steam must always develop moisture. Since all the ther- 
mal effect of the cylinder-walls must be to produce con- 
siderable moisture before expansion begins, and since 
comparatively little reevaporation takes place before the 
exhaust-valve opens, there is always a considerable per- 
centage of water thrown into the receiver from the high- 
pressure cylinder. 

In many engines, particularly where the importance of 
constructive simplicity outweighs thermodynamic consid- 
erations, the presence of this water is entirely neglected. 
In the greater number of large stationary engines, how- 
ever, the receivers are drained of this water with more or 
less care, by traps or by hand. In this way the receiver 
acts as a separator upon the passing steam ; but as scarcely 
any separator can be expected to deliver steam containing 
less than one or two per cent of moisture, on the average, 
there is still an appreciable amount of moisture which finds 
its way into the low-pressure cylinder, where it operates to 
exaggerate condensation in the manner already described. 



THE COMPOUND STEAM-ENGINE 269 

The thermodynamic disadvantages of this condensation 
and the corresponding advantages of using perfectly dry, 
or even superheated, steam, coupled with the facility with 
which such moisture can be dried out by means of coils of 
pipe filled with boiler-steam inserted in the receiver-volume, 
has led to the wide adoption of such a plan in engines of 
considerable size which aim at the best thermodynamic 
efficiency. Since the temperature within the coils is much 
higher than that of receiver-saturation the conduction of 
heat to the steam is comparatively rapid ; at any rate, until 
all moisture has disappeared. The cost of the engine is 
slightly greater with the added coils, but that of operation 
is decreased. The trap which would have been used to 
drain the receiver can be used to drain the coils. The 
receiver-steam always enters the low-pressure cylinder dry ; 
sometimes it is considerably superheated. The gain in 
efficiency is always marked. It varies so widely, however, 
with variations of service, size of receiver and coils, etc., 
that no rule can be given, either for the volumes or heat- 
ing-surface to be chosen or for results to be expected. 

The Steam-turbine. — The applicability of thermody- 
namic theory to the action of steam-turbines is very limited. 
This action is, theoretically, upon the Rankine cycle. It 
is often, and most profitably, modified by superheating the 
steam. In this case, however, arises the difference between 
the turbine and the piston-engine that in the latter the 
superheat seldom succeeds in penetrating the cylinder 
until cut-off occurs, and in so influencing expansion ; while 
in the former, since reciprocation and its corollary, con- 
densation and reevaporation, are wanting, the steam 
actually operative within the engine is superheated. 

The essential process of the steam-turbine is the pre- 
liminary conversion of the steam-heat into kinetic energy 
of translation before its absorption by the machine. If 



270 THE THERMODYNAMICS OF HEAT-ENGINES 

this process occupy the entire temperature-fall available, 
the resultant velocity of translation is excessively high ; 
and since the velocity of the vane which is to absorb its 
energy should be a fair fraction thereof, this calls for 
excessively high speeds of rotation. To avoid this the 
temperature fall is usually split up, or compounded, into a 
great many steps, each of which develops only a moderate 
velocity, which velocity is absorbed by moving vanes before 
the next step is undertaken. 

But velocity cannot be developed without concurrent 
friction. The adiabatic conversion of the heat into kinetic 
energy is therefore no sooner undertaken than frictional 
losses enter, tending to divert the path of fall from the 
vertical adiabatic toward the constant-heat curve (Fig. 14). 
The besetting sin of the piston-engine, on the other hand, 
it may be noted, is the departure of the path of fall from 
the adiabatic in the other direction, into the southwest 
quadrant, by condensation. 

The questions arising as to the proper limits for the 
development of velocity before the frictional losses become 
too great, and as to the best form of vane, of wheel, etc., 
for absorbing the velocity developed, are problems in the 
mechanics of fluids rather than in thermodynamics. 



CHAPTER III 
THE OTTO GAS-ENGINE 

The variation of the actual cycle of the Otto-type of gas- 
engine from the theoretic is much more difficult to trace 
than is that of the steam-engine. More observations of 
a much more difficult character go to the organization of a 
gas-engine test than to that of a steam-engine test. The 
data which are most essential to the complete thermody- 
namic analysis of the processes occurring within the cylinder 
cannot be gotten at all, by any means yet adopted by the 
profession ; assumptions and estimates based upon what 
observations it is possible to make must be substituted there- 
for. For instance, one of the most important observations 
needed, that of the temperature of the uncompressed charge 
of explosive mixture in the cylinder at the beginning of the 
compression-stroke, has never been taken but once or twice 
and then at a great cost of money and effort. 

The prime factors in the modification of the actual gas- 
engine cycle from the theoretic are the following. They 
are named about in the order of their importance. 

(i) The abstraction of the heat from the burning gases 
by the water-jacket. 

(2) Heat-interchanges with the cylinder-walls similar to 
the condensation and reevaporation of the steam-engine. 

(3) The influence of variations from the normal cycle 
introduced in order to modify the power developed. 

(4) The slowness of inflammation. 

(5) Heat-abstraction by the water-jacket during com- 
pression. 

271 



272 



THE THERMODYNAMICS OF HEAT-ENGINES 



Of these, numbers (1), (3), and (4) only are visible in the 
indicator-card. The others become graphically visible only 
in the entropy-temperature diagram. They can all of them, 
of course, be calculated algebraically. 

In Fig. 46 let the area BCDE represent the theoretic 
possibilities of an Otto gas-engine in the same way as did 
D ; Fig. 21, page 140. If the cor- 
responding pressure-volume dia- 
gram, Fig. 22, were reproduced 
here and the indicator-card from 
the actual engine superimposed 
upon it, there would be found 
wide discrepancies between the 
two. It is not worth while to 
do this, however, because a com- 
parison between theoretic and 
real pressure-volume diagrams 
does not illustrate differences in 
heat-energy, which is the thing 
needful, but only differences in 
their effects; and as analysis is 
undertaken only for the purpose 
of detecting causes, recourse 
must be had to the entropy-tem- 
perature diagram. The method 
of doing this mathematically will 
be explained on page 281. In 
the meantime let attention be 
turned to what happens within the cylinder. 

During the process of compression, which theoretically 
passes adiabatically from B to C, heat is abstracted by the 
water-jacket. The actual path of compression therefore 
departs to the left and follows some course similar to Be. 
Then ignition and inflammation take place and temperature 




Fig. 46 



THE OTTO GAS-ENGINE 



273 



and entropy increase along cd. The heat supplied is suf- 
ficient to carry the explosive development of heat to D, 
While the process is in operation, however, the water- 
jacket steals some 35% to 50% of the total heat supplied, or 
the area GdDH '; the isomorphic development of heat there- 
fore cannot pass beyond d. By observation of the heat 
absorbed by the jacket-water during the test of the gas- 
engine in question the point d may be determined on the 
theoretic diagram. But the abstraction of heat by the water- 
jacket does not occur in such a way as to take the simple 
form GdDH, leaving the actual cycle to be BCde. Some 
of it occurs during compression, and is measured by OBcL. 



p 


/ 


oV 


*»_* 












c 




1 




^^TO 


=Ka»t 






A- 






___ *|____ 














/ 

















i 






V 



Fig. 47 

But the greater portion occurs during explosion and the 
first portion of expansion, and is measured by HDfgheG, 
in which the line eG now takes such position as will make 
this area and OBcL together equal the measured loss of 
heat. For this reason the limit of explosive rise of tempera- 
ture,/, never quite coincides with the point d. 

Standard gas-engine diagrams may be divided into two 
sorts or classes, one from each of which is shown in Figs. 46 
and 47. The first is the true standard card of maximum 
efficiency under a given degree of compression and quality 
of gas; it is shown by BcfgheB. The other is what very 
often appears in its place, but which is usually due to no 
more serious defect than a lack of proper adjustment. It is 



274 THE THERMODYNAMICS OF HEAT-ENGINES 

shown by BcklmnB. In it inflammation takes place too 
late or too slowly. It is always due to one of two causes : 

(i) Late ignition; 

(2) Mixture of gas and air in improper proportions. 
As to its effect upon the efficiency, the pressure-volume 
diagram reveals nothing certain ; because it is not obvious, 
from a single diagram such as cklmn, Fig. 47, whether the 
area developed would be greater or less than that which 
would otherwise have been produced had inflammation been 
more prompt, such as cfgh. Or, if comparison by meas- 
urement be instituted, there is no argument whereby the 
result can be known to apply to other cases. But in 
Fig. 46 it is plain that any departure of the piston upon 
its course before inflammation is complete must result in 
work being done and hence in the bending of the curve kf 
around to the right toward first the horizontal isothermal 
and beyond that toward the vertical adiabatic, into some 
such path as klm. This is fatal to the attainment of the 
best efficiency. It is evident without further inspection 
that nothing which the cycle may do later in its course 
may hope to make good this loss of availability for work. 

The transition from the isomorphic cf to the approxi- 
mated vertical adiabatic of expansion (Fig. 46) is usually 
and properly in the form of a quite acute cusp, as at f. 
Otherwise it is more rounded, as along kl. Sooner or 
later, however, the transition is made. It is then to be 
particularly noted that the change of direction always 
swings beyond the vertically downward, or true southerly, 
direction into the southwesterly quadrant. This shows 
that the development of heat within the working-substance 
is replaced by an abstraction of heat. The only exceptions 
to this rule are in occasional cards in which the first por- 
tion of the expansion-line, fg, of a sharp-pointed card is 
truly vertical. So that even these exceptions show that 



THE OTTO GAS-ENGINE 275 

the development of heat has come sharply to an end and 
that for the moment no heat-interchanges are taking place 
whatever. 

Considerable emphasis must be laid upon this point, be- 
cause for many years much has been said and written about 
" delayed combustion " or " afterburning," and about disso- 
ciation as one of its causes. Since the basis for all such 
arguments has always been the form of the expansion- 
curve, it is important to note that the first thing proved by 
the form of that curve is that combustion is always com- 
plete by the time that maximum pressure is reached ; or 
that if it be delayed it is only in those cases where the 
incipience of combustion has been delayed. In other 
words, explosive combustion in an Otto gas-engine, instead 
of being a dual or complex process, part of it occurring 
immediately after ignition and part of it delayed by some 
mysterious cause until late in the expansion-stroke, appears, 
when seen in the entropy-diagram, to be a simple continu- 
ous process occupying a fairly definite amount of time. If 
it be inaugurated well before the crank reaches dead-centre, 
it is usually complete by the time the piston has moved 
appreciably on its outward stroke. If it be not started 
until that stroke has begun, or if the mixture be a slow 
one, it may last continuously until the stroke is well toward 
completion. But there is no sign of its being interrupted, 
or of its being delayed or postponed by any other causes 
than those which always lead to the lapse of an appreciable 
period of time in explosive combustion. 

It is next to be noticed that the portion of the cycle just 
under discussion, which is always convex toward the right, 
whether in the form of an apparently sharp point, as at f, 
or of a well-rounded curve, such as kl, is always followed 
by a process which is concave toward the right. The only 
exceptions to this rule which the writer has ever met are in 



276 THE THERMODYNAMICS OF HEAT-ENGINES 

those cases where the exhaust-valve opens before combus- 
tion is, or could reasonably be expected to be, complete. 

The significance of this change of curvature is marked. 
Convexity toward the right shows the rate of heat-supply 
becoming steadily less. Mathematically, it might also show 
rate of heat-abstraction steadily increasing ; but considera- 
tion of conditions actually prevailing makes this proposition 
untenable. 

Concavity toward the right, on the other hand, necessa- 
rily shows active development of heat. Where the con- 
cavity has not yet brought the expansion-curve again into 
the vertical adiabatic there may be some debate of this 
proposition. Where it has brought the curve into the south- 
easterly quadrant, however, — and this is nearly always 
the case, — there can be no question. The departure of 
the curve to the east shows the addition of heat. 

It was this portion of the process which led to the theory 
of delayed combustion. Because the pressure-volume dia- 
gram is very meagre in its information, it was not observed 
at the same time that this addition of heat is separated 
from the original development of the fuel-heat of the gas 
•in combustion by a distinct period when no production of 
heat took place. This intervening period being now known, 
what explanation of the situation is most probable ? 

Since the expansion-curve, whether of the form of fgh 
or of klmn, is strikingly similar to the expansion-curve of 
the simple, non-jacketed steam-engine, 1 and since in the 
latter case there is no possible source of heat except the 
cylinder-walls, it is only rational to assume that there takes 
place in the gas-engine cylinder a reversing transfer of heat 
between working-substance and walls which is in every way 
the parallel of the familiar condensation and reevaporation 
of the steam-engine cylinder. Such a simple explanation 

1 See Fig. 41, page 219. 



THE OTTO GAS-ENGINE 277 

covers the demands of all the known facts. It eliminates 
the unscientific mystery of " delayed combustion." It may 
be assumed as demonstrated that nothing occurs within the 
gas-engine between the point of ignition and the opening 
of the exhaust-valve but the prompt, continuous combus- 
tion of the charge, its expansion due to the motion of the 
piston, and the varying action of the jacketed walls upon 
it in the ways naturally to be expected. 

It is a widespread fault in the commercial and profes- 
sional treatment of the gas-engine which has led to this 
doubt and confusion ; and against it the student should be 
warned. I refer to the common practice of making re- 
peated tests of gas-engines at great expense of time, labor, 
and money, without commensurate effort at an analysis 
of the results obtained. It is of little use to ascertain the 
facts unless we learn their causes. Such knowledge of 
causes cannot be had from the indicator-card direct. The 
compression and expansion curves of Fig. 49 are dumb as 
to all the thermal phenomena of which their translations 
into the curves of Fig. 48 become eloquent. Such trans- 
lation is simple and fairly expeditious. It is not to be 
understood why it is not more frequently performed. Its 
omission is in line with the frequent elaborate reports of 
elaborate tests, with no mention made of the clearance of 
the cylinders or of the calorific power of the gas. 

The Entropy-temperature Analysis. — In the translation 
of pressure-volume or indicator diagrams into entropy-tem- 
perature curves, the following argument is relied upon : — 

Let Fig. 49 be one of the indicator-diagrams, supposedly 
an average one, of a set taken under constant conditions. 
Let Fig. 48 be an entropy-temperature coordinate field on 
which has been located only the initial point B. 

In the location of this point B only the temperature need 
be considered. As to entropies, all that is usually desired 



278 THE THERMODYNAMICS OF HEAT-ENGINES 

in the entire analysis is a knowledge of comparative altera- 
tions, to know whether heat be entering or leaving the 
working-substance. But even where the quantities of heat 
leaving or entering are desired, and where entropies are 
therefore to be taken to exact scale, the zero-point may be 
located horizontally arbitrarily. As to temperature, how- 
ever, there is considerable room for discussion. It cannot 
be known by observation, except at a cost which is nearly 
always prohibitive. Without it the absolute temperature 
at any point in the cycle cannot be known. But, on the 
one hand, much can be learned from the entropy-analysis 
with only knowledge as to comparative alterations of tem- 
perature. On the other, the temperature B cannot be out- 
side of certain fairly definite limits. It cannot be below 
atmospheric temperature. The engine cannot have raised 
it above atmospheric temperature by more than a reason- 
able amount. Finally, since the temperatures after explo- 
sion, as calculated from the assumed initial at B, will be in 
error by four or five degrees for every one of error in the 
assumption at B, too great error will be checked out by 
improbability at either the top or the bottom of the cycle. 
The writer usually assumes the temperature of B at the 
convenient round number of 6oo° absolute, Fahrenheit. 1 

On the indicator-diagram first lay off the vertical axis 
OP (Fig. 49) at the proper distance to the left to measure 

1 Professor Burstall's measurements show higher temperatures, at all por- 
tions of the cycle, than would justify this assumption. The mean of those made 
for the Gas-engine Research Committee of the Brit. Inst. M. E., upwards of 
250 F. during suction and 3600 F. during explosion, would point to an 
assumed initial temperature properly as high as 700 absolute, instead of 6oo°. 
But as these measurements were made with a very small engine, running below 
speed and missing some five explosions out of six, the rigidity with which they 
may be applied under more normal conditions must be questioned. The 
writer's figure, 6oo°, is undoubtedly as low as the truth. (See Trans. Brit. 
Inst. M. E., 1901.) 



THE OTTO GAS-ENGINE 



279 



the clearance. Also draw the zero-axis of pressure, O V. 
Then erect any number of ordinates through any desired 
points whatever. It will save work to carry these ordi- 
nates completely across both curves. Then in order to 
locate upon Fig. 48 any desired point, such as x, of Fig. 
49, lay off (in the imagination) from B, Fig. 48, a constant- 
pressure isomorphic By. This must represent geometri- 
cally the same process as By of Fig. 49. Whether any 




Fig. 48 



such process actually took place or not does not affect the 
mathematical argument. From y, Fig. 48, lay off the con- 
stant-volume isomorphic yxz. This must correspond in 
the same way to the line yxz of Fig. 49, and upon it may 
be located both x and z. 

The temperatures and entropies of the points y, x, and 
z may be had by calculation by remembering — 

(1) That on the constant-pressure process By the tem- 
peratures must be proportionate to the volumes, which are 
measurable from OP ; 



28o 



THE THERMODYNAMICS OF HEAT-ENGINES 



(2) That on the constant-volume process yxz the tem- 
peratures must be proportionate to the absolute pressures, 
which are measurable from O V; and 

PV P V 

(3) That in any case the equation — — = — }—± holds true. 

T 7\ 

Let the pressures and volumes at the points B, x, and z 
be measured in any convenient scale whatever ; a decimal 
scale will be found the most convenient. Let the ratios 

— -, — , etc., the ratios — -, — -, etc., and the ratios — -, —*-. 

v;vj p;p b ' pjp; 

etc., be found and tabulated. (By setting a slide-rule to, 

first, V B , and then to P B , and going over the entire diagram, 

p 




Fig. 49 



first for volume-ratios and then for pressure-ratios, this can 
be done very expeditiously.) 

Next, note for each desired point the logarithms of these 
ratios. If greater accuracy be desired, the logarithms of 
the original measurements might have been noted, the 
logarithms of P B or V B being subtracted from them ; but 
this is seldom necessary. 

From the equation noted in paragraph (3) above : — 



log ^ = log ^- log -p. 



(See page 120.) 



From page 83, 



N y = S v log, i and N B - N y = S p log e Sj 

J-y 1 y 



THE OTTO GAS-ENGINE 28 1 

Therefore, 

N X -N B = S v log e ^-S p log, |> 

-* y ■*■ y 



= S v flo Se ^- 1.408 log e p) 

= 2.3026 >s; (log -? - 1.408 log — ^] 

\ J y 1 yj 

= 2.3026 SJlog -^- 1.408 log-^J. 



When the actual scale of heat-interchanges is desired, the 
value of the coefficient 2.3026 S v must be determined, the 
mass of the gas involved being carefully considered ; or, 
more simply, the whole calculation may be considered as 
covering that number of working-strokes which will bring 
the value of 2.3026 S v times the mass of the gases passing 
through the engine in that period equal to unity. Usually, 
however, entropy-values to any scale will suffice, in which 
case the coefficient 2.3026 S v may be neglected. 

In either of the latter two cases the calculation for any 
given point, such as x, reduces to this simple form : — 

^=1.490 log -^ = 0.1732 

Adding 0.4 1093 

and then 0.008 22 

0.2847 

^=1.778 log ^ = 0.2499 

Subtracting, ^—^=—0.0348 

^=3.602 log ^ = 0.5565 



p 

Adding log -~ to N x — N B above : — 



N,-N B = o.$2i7 



282 THE THERMODYNAMICS OF HEAT-ENGINES 

V P T 

Subtracting log — y from log -f-, log — - = 0.0767 

log T B = log 6oo° = 2.7781 

Adding, log T x =2.8548 

T x = 715-8 

4 = 255° F. 
p 
Adding log — z - to log T x : log T z = 3.41 1 3 

■* X 

T z = 2578 

4 = 2117° F. 

This gives the entropies relatively to B and the absolute 
temperatures of any desired pair of points. If entropies 
be desired to true scale, they may now be multiplied by 
2.3026 S v times the mass of gas passing through the 
engine per stroke or per unit of time. 

The entropy-diagram of the actual cycle thus obtained 
may then be analyzed in a manner very similar to that 
given for the steam-engine. In such analysis, as in every 
portion of the test of which it forms a part, the threefold 
object of the work should be kept clearly in mind, viz. : — 

(1) The determination of the division of the heat sup- 

plied in the fuel between the three destinations : — 

(a) Absorption by the jacket-water; 

(b) Conversion into work ; 

(c) Rejection in the exhaust ; 

(2) The comparison of these results with the theoretic 

possibilities ; 

(3) A clear understanding as to the causes of the results 

observed. 

The observations which must be taken in order to com- 
plete this programme are : — 



THE OTTO GAS-ENGINE 283 

(1) For determination of the heat supplied : 

(a) A gas-meter, protected by. a gas-bag and prop- 

erly calibrated ; 

(b) A gas-calorimeter, preferably a Junkers ; 

(c) An admission-counter, operated by the admis- 

sion-valve. 1 

(2) For determination of the conversion of heat into work : 
{a) The indicator, equipped with two springs for use 

as noted below ; 

(b) A Prony brake or absorption-dynamometer; 

(c) A revolution-counter ; 

(d) An explosion-counter. 

(3) For determination of the heat lost to the jacket-water : 
(a) The rate of flow, which should be kept constant 

during and for some time preceding the test ; 
{b) The temperature at entrance to the jacket; 
(c) The temperature at exit from the jacket. 

(4) General data : 

(a) The barometric pressure ; 

(b) The temperature of the air entering the suction- 

pipe ; 

(c) The temperature of the exhaust-gases (to be 

attempted only when the engine is running 
at maximum power, missing no possible ex- 
plosions); 

(d) Chemical analysis of the exhaust-gases, including 

observation of the presence of uncombined 
carbon. 

(e) The dimensions of the engine, including the vol- 

ume of the clearance. 

1 Where the igniter is suspected of uncertainty of operation this instrument 
should be supplemented by an explosion-counter. 



284 THE THERMODYNAMICS OF HEAT-ENGINES 

As to the taking of indicator-cards, two sets should be 
secured. One should be with the standard gas-engine indi- 
cator, with a spring heavy enough to keep the vibratory 
waves within proper limit. This will be used as already 
noted. The other set should be taken with a standard 
steam-engine indicator, with a spring so light that the com- 
pression of the charge brings the indicator-piston to, or quite 
near to, the top of its stroke. Such diagrams show the phe- 
nomena occurring during the exhaust, suction and compres- 
sion strokes upon an exaggerated scale, with much greater 
accuracy than the former ones. Those, indeed, should be 
relied upon for the explosion and expansion curves only. 

The diagrams from the light spring will also reveal the 
negative work involved in overcoming the fluid-friction of 
admission and exhaust. This should be carefully distin- 
guished from the metallic friction of the engine, in making 
the report and drawing conclusions. It is usually high in 
gas-engines, apparently more so than is necessary. 

If the spring used for the second set be too strong to let 
the compression bring the indicator-piston nearly against 
its upper stop before the explosion, the latter, when it oc- 
curs, will wreck the pencil-motion. 

As to the explosion-counter, that is most easily rigged by 
using a second indicator which is always kept open to the 
engine-cylinder and the drum of which is given a slow con- 
tinuous or ratchet motion by the engine just sufficient to 
cover a millimeter or so for each cycle. Its record will con- 
sist of a number of saw-teeth, high ones recording explosions 
and low ones signifying missed explosions. Such a record 
will check out any failure to ignite charges of gas which the 
admission-counter has recorded as entering the engine. 

If a second indicator be not available, the reliability of 
the igniter may be tested in this way before and after 
the regular test. 



THE OTTO GAS-ENGINE 285 

The gas-engine test made in this way should result in a 
report recording : — 

(1) The dimensions of the engine and the conditions of 

the test ; 

(2) The observations themselves, reduced to averages 

and totals ; 

(3) The calculated distribution of the heat to its four 

destinations, in percentages of the whole ; 

(4) The four calculated efficiencies, viz. : — 

(a) The theoretic or ideal efficiency, calculated from 

the volume-ratio of compression ; 

(b) The thermodynamic efficiency, which is the same 

as the second item of paragraph (3), this page ; 

(c) The cylinder-efficiency, or (d) divided by {a) ; 

(d) The mechanical efficiency ; 

(5) Average samples of cycle-diagrams, both from the 

indicator and in entropy-temperature coordinates ; 

(6) The conclusions to be drawn from the foregoing. 

(By the time this book has left the press it is probable 
that the committee of the Am. Soc. M. E. which was ap- 
pointed to adopt standard methods of making and record- 
ing steam-engine and gas-engine tests will have made its 
final report. This report will be available to most students 
and should be consulted before undertaking a test.) 

Degree of Compression 

In the study of the attainment of maximum actual effi- 
ciency in the Otto gas-engine it would seem at first that 
the problem were a very simple one. The actual efficiency, 
though less than the theoretic, is very closely proportional 
to it. The latter is given by the expression 

T — T f V\°- m 

E = ±3 ^ = 1 -fi-3 1 



T t \VJ 



286 THE THERMODYNAMICS OF HEAT-ENGINES 

wherein the subscripts i and 2 refer to the conditions at the 
beginning and end of the compression-stroke respectively. 
It is obviously easy, in the design of the engine, to make 
this expression take any value we choose, by variation in 
the proportion of clearance-volume. 

In the development of the engine during the quarter- 
century which has elapsed since its first appearance, this 
policy of increasing the efficiency by decreasing the clear- 
ance has been steadily pursued. It has constituted the 
main line of progress up to about 1895. Since these 
thermodynamic principles were well known from the start, 
it is difficult to see at first why advantage of them was 
taken so slowly. The explanation is that the mechanical 
difficulties of construction multiply very rapidly as the 
degree of compression increases. The maximum pressure 
incurred at the moment of explosion rises very rapidly 
with a rise in initial compression. With it must rise the 
entire cost of construction of the engine ; all parts must be 
strengthened and weighted and bearing-surfaces must be 
increased. Indeed, so close has constructive practice fol- 
lowed mechanical limitations in this matter that nothing 
has permitted the later advances into the higher degrees 
of compression but the steady rise in rotative and piston 
speeds which has also been a prime characteristic of prog- 
ress, in gas-engines as in steam-engines. At the higher 
speeds the force needed to accelerate the reciprocating 
parts subtracts itself from the fluid-pressure on the piston 
and protects the bearing-surfaces from momentary over- 
pressure. It is to be noticed at the same time, however, 
that such inertia-forces fall upon the bearing-surfaces un- 
modified by fluid-forces during the suction and exhaust 
strokes ; they may not be increased recklessly, therefore. 

Another limit to the feasible degree of compression is 
found in the danger of pre-igniting the charge by the heat 



THE OTTO GAS-ENGINE 287 

of compression. This limit appears earlier in oil-engines 
than in gas-engines. It is a phenomenon which may be 
taken advantage of, however. All engines relying upon 
hot-tube ignition without the aid of timing-valves depend 
upon it, more or less unconsciously, for ignition. The 
Diesel engine relies upon it absolutely and completely ; 
adopting a maximum pressure of compression of from 500 
to 550 pounds per square inch, it insures at dead-centre an 
atmosphere within the cylinder all portions of which are 
above the temperature of ignition. Ignition is timed in this 
case by the preservation of the fuel separate from the air 
until the beginning of the working-stroke. It is then fed 
into the hot compressed air under control and ignites because 
of the high temperature of the air. The most valuable re- 
sult yet derived from the Diesel engine is the demonstration 
that in such an atmosphere as this any sort of liquid fuel, 
whether light or heavy, crude or pure, may be injected 
without preliminary vaporization and the resultant combus- 
tion will be complete. 1 Referring again to the analysis of 
the theory of the Diesel cycle, which was presented on 
page 149, it is to be noted that its high efficiency is due 
primarily to its high degree of compression rather than to 
any peculiarities of form of cycle, method of ignition, " iso- 
thermal combustion," etc. 

Complete Expansion. — The next line of effort at improve- 
ment of efficiency, taken in the chronological order of their 
appearance, was the avoidance of incomplete expansion 
and incomplete exhaust of the burned gases. These fea- 
tures, it will be remembered, are inseparable from an Otto- 
cycle engine in which all of the processes are enacted in a 
single cylinder. They are now accepted as such, but only 
after much time and money and some of the best skill in 

1 " Mittheilungen iiber den Dieselschen, Warmemotor," by Rudolf Diesel. 
Zeitschrift d. Vereines Deutsche Ingenieure, Vol. XXXXIII. 



288 THE THERMODYNAMICS OF HEAT-ENGINES 

the land has been expended in proving the impracticability 
of their avoidance. Chief amongst the workers in this line 
was Mr. Atkinson, who made two full attempts at design- 
ing an engine which should completely expand and exhaust 
its burnt gases. In the first a horizontal cylinder open at 
both ends contained two trunk-pistons placed head to head. 
Each of these was connected to the engine-crank by a 
peculiar arrangement of links and walking-beams whereby 
each piston had a peculiar and independent motion of its 
own. Considering first the relative motion between the 
two pistons, it may be described as producing the following 
sequence of events : — 

(i) The close juxtaposition of the two, which constituted 
the beginning of the suction-stroke ; 

(2) Their separation by a certain distance, whereby the 

volume between them was filled with an explosive 
charge at atmospheric pressure ; 

(3) Their approach by a lesser distance, whereby com- 

pression was performed, at the end of which pro- 
cess ignition took place ; 

(4) Their separation by a distance greater than (2), 

whereby expansion took place much more com- 
pletely than in the ordinary engine ; 

(5) Their close approach, into position (1), whereby com- 

plete exhaust was effected. 

This engine was known as Atkinson's " Differential " 
engine. Not proving a mechanical success, it was fol- 
lowed by his so-called " Cycle " engine, in which the same 
series of changes of volume was produced in a cylinder 
closed at one end and containing a single trunk-piston, by 
a peculiar radial gear between piston and crank. Both 
engines gave good thermodynamic results. Both were 
complete failures mechanically. The work was done so 



THE OTTO GAS-ENGINE 289 

well, however, that the failures are now regarded as inev- 
itable in that line of effort. 

Scavenging. — Although the completion of expansion in 
the Otto-type engine still remains an unsolved problem, 
the completion of exhaust has been accomplished in a 
simpler way. For a time a number of designers advocated 
the insertion between each two-revolution cycle of one 
complete idle revolution, the end in view being the pump- 
ing of a fresh charge of air through the engine, whereby 
the bulk of the burnt gases remaining in the clearance- 
volume should be swept out. But the gain was not found 
to be worth the mechanical and commercial disadvantages 
incurred, and the plan is now no longer followed. Later 
Mr. Atkinson evolved the beautifully simple plan of pro- 
ducing a partial vacuum in the cylinder during the exhaust- 
stroke by making the exhaust-pipe some sixty feet long or 
more, and quite straight and free. The inertia of this long 
column of gases, when once set into rapid motion by the 
first sharp puff of the exhaust, would draw into and through 
the cylinder quite a draft of pure air, washing out the bulk 
of the burnt gases in the clearance. The thermodynamic 
effect of this plan was excellent. His early experiments 
with it reported a saving of some 5%, a quite profitable 
one considering the ease with which it is attained. 

Scavenging, in some form or other, is now considered an 
essential feature of all large explosive engines attempting 
refined efficiency. In the latest types of heavy engines for 
use with blast-furnace gases it is performed by a small 
separate blowing-cylinder. 1 

Regulation. — Of all the problems connected with the 
design of the Otto-type of gas-engine, however, none has 

1 See Humphrey's paper on " Recent Progress in Large Gas-engines," read 
before the British Association at Belfast, Sept. 11, 1902. Also abstracted in 
London Engineering, Sept. 19 et seq. 
U 



29O THE THERMODYNAMICS OF HEAT-ENGINES 

a more urgent bearing upon the thermodynamics of the 
efficiency-problem than that of regulation. It will be re- 
membered that all of the early designs operated upon what 
is called, most properly, the " hit-or-miss " plan of regula- 
tion. That is, so long as the engine were below normal 
speed there was no hindrance to, nor modification of, the 
development of full maximum power. When the speed 
rose above normal, all development of power was shut off 
completely by the omission of gas-supply on one or more 
of the suction-strokes. There was no gradation between 
these two extremes. 

Many of the smaller engines, and a few of the larger 
ones, still adhere to this plan. When it is remembered 
that the single-cylinder gas-engine has three idle strokes 
to one working stroke, at the best, and that the dropping 
out of one working stroke leaves seven idle strokes before 
another impulse can be received ; when it is further remem- 
bered that this seven-to-one ratio applies to a reduction of 
power to only half the maximum, or about seven-tenths the 
proper rating, and that for further reductions the propor- 
tion of idle strokes to impulses may be increased to eleven, 
fifteen, or nineteen to one ; when it is finally remembered 
that the power needed for compressing the next charge to 
be fired must be taken out of the fly-wheels at just the 
moment when they are the weakest and after they have 
already fallen below normal speed ; when all of these points 
are compared with modern demands and attainments in 
steam-engine regulation, the hit-or-miss plan of regulating 
gas-engines must appear as a crudity only comparable with 
steam-engine practice of a century ago. 

Nor can any remedy possibly be hoped for in the gov- 
ernor. The above remarks would apply with complete 
force were the governor ideally perfect. It is true that in 
many small gas-engines the governor is abnormally and 



THE OTTO GAS-ENGINE 29 1 

inexcusably bad, and that the speed-variations are thereby 
much exaggerated over what is inevitable. In fact, no gas- 
engine operating upon this plan is fitted with a good gov- 
ernor. The governor is always one of two types, both of 
which are intrinsically unsuited to the work, viz.: (1) A 
very small and weak centrifugal governor, dimensioned so 
as to have great stability ; (2) an inertia-governor, usually 
of the pendulum-type. 

The latter is unsuited because its action depends upon 
acceleration ; it can do nothing if not to keep the engine's 
acceleration, positive or negative, within certain limits. 
But the Otto-type of gas-engine necessarily involves a 
degree of acceleration within each cycle, positive during 
the expansion-stroke and negative during the compression- 
stroke, which far exceeds any degree of general acceleration 
in average speed which is permissible in good regulation. 

The former type is unsuited because the range of speed- 
variation which is involved in stability and which consti- 
tutes a valuable factor for control in steam-engine governors 
is here entirely unavailing. The governor can do nothing 
but one extreme thing or the other, either to admit maxi- 
mum power or cut off all of it, and a field of gradation 
between is powerless to accomplish gradation. Moreover, 
owing to these extremes in distribution of power, positive 
acceleration will be prompt and powerful ; in addition, 
from one to four strokes must elapse before action on the 
part of the governor can become effective. The need is, 
therefore, for a governor which will signalize departure 
from normal speed with the utmost promptness and em- 
phasis. Such is the perfectly isochronous pendulum gov- 
ernor, a type entirely unfit for steam-engine regulation. It 
is the only one fit for the hit-or-miss plan of controlling 
gas-engine speeds. It has never, so far as the writer's 
knowledge extends, been applied. 



292 THE THERMODYNAMICS OF HEAT-ENGINES 

It is no object of the present text to properly discuss 
questions of regulation other than as they affect thermo- 
dynamic problems. But the latter cannot be correctly 
entered upon without previous acceptance of the premises 
that the hit-or-miss plan of governing is intolerable, accord- 
ing to all steam-engine standards of excellence, and must 
inevitably remain so. This being true, what other plans 
are available ? Two only, viz. : — 

(i) Reduction of amount of gas present at the moment 

of ignition. 
(2) Reduction of the pressure prevailing at the moment 

of ignition, by the reduction of either the volume 

(by cut-off) or the pressure (by throttling) of the 

charge drawn in during suction. 

Either (1) or (2) permits the gradual reduction of the 
vigor of each impulse, without diminishing their number, 
until only a small fraction of normal power is developed. 
Either plan inevitably involves some loss of efficiency from 
the maximum. In the case of the first plan this is due to 
the inevitably slower combustion in weak mixtures, and is 
so bad as to be intolerable. Reference to the theory of 
the Otto cycle will confirm the statement as to the second 
plan also. In the expression for its theoretic efficiency, 

T —T 

2 — 1, reduction of the degree of compression reduces 

the maximum efficiency attainable. With the second plan 
it is only a partial, but a valuable, compensation that 
decrease of the degree of compression (in a fixed clearance- 
volume) increases the ratio of expansion into that territory 
which was stated, on page 148, Part I, as belonging exclu- 
sively to the Lenoir cycle. 

The objections to both plans can be modified by skilful 
design and particularly by a wise combination of the two. 



THE OTTO GAS-ENGINE 293 

Practically all of the larger and more important engines 
have accepted the premises stated above as to the hit-or- 
miss plan and rely upon graded impulses for regulation. 

It is further to be remarked that variation of the power 
developed, with or without variation of the degree of com- 
pression prevailing at the moment of ignition, is obtainable 
by any one of three methods, viz. : — 

( 1 ) Gradation of pressure of charge during suction, or by 

throttling. 

(2) Gradation of volume of atmospheric charge during 

suction, or by variable cut-off. 

(3) Gradation of final volume of compression, or by vari- 

able clearance-vohi7ne. 

The first two are very nearly alike ; their difference is 
visible in Fig. 56. The second has the advantage over 
the first by a narrow margin of efficiency and the disad- 
vantage of greater mechanical complexity. The third, so 
far as the writer is acquainted, has never been attempted. 

To illustrate these points is published herewith a set 
of indicator-cards, with entropy-analyses thereof, from 
an 11" X 12" two-cylinder gas-engine, running at 290 
revolutions per minute. The conditions under which the 
cards were taken were in every way normal and were 
graded from an overload of 17.4% above rating down 
to the friction of the engine alone. The fuel-consump- 
tion near rated load varies between 10 and 10.6 cubic feet 
of natural gas per brake horse-power, the gas having 
an average calorific power of 1000 B.t.u. per cubic foot. 
This represents a thermodynamic efficiency of from 24 to 
25.5%. The variation of the fuel-rate with the load is 
remarkably constant. Calling the efficiency at rated load 
100, the indicator-cards show it as rising to about 112 
under the overload and falling to about 75 at half -load. 



294 



THE THERMODYNAMICS OF HEAT-ENGINES 



This is accomplished, so far as the cards reveal the 
method, by relying upon more or less throttling of the in- 
coming charge under all conditions of load, even the maxi- 
mum, and by thereby securing an expansion to a volume 
greater than the initial when the latter is measured at 
atmospheric pressure. The percentage-fraction of the 
stroke at which the charge is at atmospheric pressure is 
shown by the figures beneath the atmospheric line. It 
will be seen that only 71% of a cylinderful of fresh 
charge is taken in even under maximum load. At normal 




Rated Load 

Fig. 50 



load it is only 68%. At the same time, the clearance is 
made less than usual (only about 21.5%) so that the proper 
degree of compression is retained. The reduced pressure 
during suction, due to the throttling by the governor, and 
the work lost thereby, could be shown (without confusion 
of lines) only in Fig. 56, and there only approximately. 

While the governor is reducing the power by throttling 
the charge, it is also reducing the proportion of gas present, 
making the inflammation slower, which is quite permissible 
in view of the excessive opportunity for expansion present 
during the lighter loads. 

As these diagrams are typical representatives of the best 



THE OTTO GAS-ENGINE 



295 



practice of the day with large engines, it is proper to use 
them as a text for a discussion of the possibilities for further 
progress, as revealed by thermodynamic considerations. 

In the first place, the figures given above for the varia- 
tion of efficiency in terms of variation of load are mislead- 
ing, in that the efficiency at rated load was taken as the 

230a 




Bated Load. 

Fig. 51 

standard. It is one of the marked distinctions between 
steam-engine and gas-engine practice that, whereas in 
steam-engines (except in the almost obsolete throttle- 
governed type) the maximum efficiency is found at or 
slightly below rated load, decreasing as the load either 
increases above or falls to a fraction of the rating, in gas- 
engines the efficiency is always the greatest when the load 
is the greatest. This fact led to the very unfortunate 
practice, which is only now just dying out, of rating gas- 



296 



THE THERMODYNAMICS OF HEAT-ENGINES 



engines so high that they could seldom be driven appreci- 
ably above their rating, and often not fully up to it. Even 
the much more liberal practice shown in the accompany- 
ing diagrams reveals a maximum load only 18% above 
the rating ; and, while the writer does not know that this 
is the extreme maximum capacity of the engine, it is plain, 
from the degree of compression incurred at this power, 
that it could not possibly rise much higher. This margin 
above rating has been obtained, too, by artificially adopting, 



D 390 




Overload: 17.4$ above Rated Brake Power 



Fig. 52 

as the standard efficiency of the engine, an efficiency which 
is only some 78% of what the engine plainly shows that it 
can do when given a chance. Yet, after all, the margin of 
power above the rating, 18%, which is purchased at the 
price of the loss of 22% of the efficiency, is very small 
when compared with standard steam-engine practice, and 
plainly reveals one of the chief reasons why the gas-engine 
fails to meet more universally the demands of the industrial 
world for a prime mover. It does not possess sufficient 
margin of power. (See curve W B , Fig. 58.) 

Does theory offer any promise of remedy for this situa- 



THE OTTO GAS-ENGINE 



297 



tion ? It does, very plainly. The remedy may not be 
easily attainable in constructive practice, but it is plainly 
there. In the writer's opinion, moreover, the reduction of 
it to practice does not involve any obstacles impossible of 
conquest. 




Overload; 17.4$ above Rated Brake Power 
FIG. S3 

The practice, as well as the theory, of the governing of 
Otto-cycle engines by throttling or cutting off the incoming 
charge shows that the maximum efficiency is obtained 
when the least throttling is done and the cycle approaches 
most nearly to the unmodified Beau-de-Rochas cycle. The 
departure from the latter is undertaken, very plainly, for 
the sake of margin of power and for better regulation, 
which are purely mechanical questions, and quite in the 
face of all considerations of efficiency. 



298 THE THERMODYNAMICS OF HEAT-ENGINES 

The efficiency of the Beau-de-Rochas cycle is given by 

T — T 
the expression 2 — 1 , wherein the T's refer to tempera- 

^2 
tures before and after compression. Therefore, very 

plainly, the degree of compression should always be kept 
constant, at the maximum point permissible from me- 
chanical considerations of bearing-surfaces and strength 
of parts or from danger of pre-ignition. At the same 
time, since any reduction of the proportion of gas in 
the fresh charge leads to a slower rate of combustion, 
which is fatal to efficiency, that should be kept constant 
too. Since any reduction of the pressure of the fresh 




Half Rated Brake Load 
FIG. 54 

charge below atmospheric leads to loss of power in re- 
compression, not to mention the much greater loss due to 
the diminished pressure at the instant of ignition which 
has been discussed above, that, too, should be avoided; 
yet it is the least objectionable in the list. 

If all of these rules are to be observed, while the power 
developed and the gas taken in are reduced to suit a 
diminished resistance, only the following policy is left open 
to the designer : — 

(1) To reduce the volume of the atmospheric charge of 

standard mixture taken in ; 

(2) To reduce the clearance-volume in like proportion, in 

order to maintain a constant degree of compres- 
sion. 



THE OTTO GAS-ENGINE 299 

The first cannot be accomplished, in the standard Otto 
engine, except by recourse to an automatic cut-off. As the 
throttling of the charge is a method only slightly inferior 
to this thermodynamically, and superior to it mechanically, 
this first point will be disposed of by leaving these alterna- 
tives open to the designer without further discussion. 



2200 




Half Rated Brake Load 
FIG. 55 

(It should be said, however, that the friction in getting 
the gases into and out of a gas-engine cylinder usually con- 
stitutes as much as two-thirds of its entire frictional loss, 
and this is much exaggerated where throttling is substi- 
tuted for automatic cut-off. The entire question of engine- 
friction is one of the most unsatisfactory ones connected 
with gas-engine design. In the present case the mechani- 
cal efficiency is between 75 and 80%, which is certainly 
much below what engineering skill ought to be able to com- 



300 THE THERMODYNAMICS OF HEAT-ENGINES 

mand. The reasons for the present faulty situation are 
two : (i) Combustion within the cylinder, demanding small 
poppet-valves for admission and exhaust ; (2) the double 
transmission of the energy required for compression, first 
from the piston through all the oblique connections and 
bearings to the fly-wheel and afterward back again. 
Neither of these may be remedied without radical depar- 
ture from existing arrangement of parts.) 

As to the second rule essential to efficiency, the varia- 
tion of clearance-volume, as this is not a book upon con- 
structive mechanics, the question as to how variation of the 
clearance-volume is to be accomplished will not be opened. 




No Brake Load: Engine-Friction Only. 

Fig. 56 

It is quite proper, however, to inquire as to what would be 
the thermodynamic results were such a construction opera- 
tively accomplished. 

For this purpose let Fig. 52 be taken as representing the 
best of modern practice in the handling of gas-engine fluids 
in an engine-cylinder. Let the new engine be arranged 
to develop, under maximum load, the facsimile of Fig. 52 
chopped off at BE, or the indicator-diagram BCDE. This 
would call for a clearance of about 30^. But this 30^ 
must be supposed to be the maximum limit of a clearance 
which is variable by the governor from 30^ down to the 
minimum practicable. 

If, now, for all lesser loads than maximum the initial 
volume of the charge AB be diminished in somewhat equal 
proportion, and the clearance-volume be diminished in exact 



THE OTTO GAS-ENGINE 3 QI 

proportion with AB, the indicator-cards developed would be 
the exact likeness of BCDE, Fig. 52, upon a diminished lon- 
gitudinal scale, and would develop a like efficiency ; except 
that now the terminal pressure EB has opportunity for fur- 
ther expansion during the remainder of the stroke, to the 
right of the new positions of E. This added expansion is 
not obtained, as is the case in all existing engines, at the 
expense of diminished efficiency of the portion BCDE, due 
to diminished compression. It is all pure gain, and serves 




JSo Brake Load. Engine Friction Onjy 
FIG. 57 

to make the efficiency an increasing instead of a decreasing 
function of the load as the latter falls from maximum. 

The movement of B toward A under the influence of the 
governor as the load diminishes would result in the vari- 
ation of efficiency with load according to the curves of 
Fig. 58. In Fig. 58 the actual diagram BCDE of Fig. 52 is 
taken in each case as a standard for comparison of 100. 
The curve W B shows the approximate relation of the brake- 
efficiency of the existing gas-engine under discussion to 
its brake-load. The curve N B shows the same relation for 



302 



THE THERMODYNAMICS OF HEAT-ENGINES 



the proposed new engine with variable clearance-volume 
controlled by the governor. The curve Nj shows the rela- 
tion of indicated efficiency to indicated power for the new 
engine. 

In considering these curves, it must be remembered that 
the curve N B is a theoretic one only in the supposition of 
altered dimensions of clearance-space. All the inevitable 
departures of actual results from theoretic possibilities are 




50 100 150 

Brake Load, in Percentage of Rating 

Fig. 58 

included in the premises. It is also to be noted that the 
curve W B represents, not poor engineering practice which 
needs rebuke, but what is possibly the best, and surely 
near to the best, of modern practice. The curve N B shows 
in comparison the undeveloped possibilities of the Otto 
engine, so far as the writer yet perceives them. 

In comparing the curves W B and N B , the point of prob- 
ably greater importance than the generally higher average 
efficiency of the latter is the wide range of its capacity. 



THE OTTO GAS-ENGINE 303 

The rating as chosen shows a surplus margin of capacity 
of 70%. It is probable that the average builder would 
prefer to rate such an engine at about 120 (on the arbitrary- 
scale adopted), reducing its margin of surplus of maximum 
above rated power to some 41%, but at the same time re- 
ducing the lower limit of fractional load which might be 
imposed without perceptible detriment to the efficiency to 
50%. The limit to reduction of power without passing 
below what is now first-class gas-engine efficiency at rated 
load would be thereby reduced to 33%. In other words, 
an engine consisting of a pair of I3"xi4" single-acting 
cylinders running at 275 r.p.m., if equipped with governor- 
controlled variable clearance, would rate at 60 horse-power 
and would develop that power at a fuel-rate of 8^ to 
9 cubic feet of natural gas per brake horse-power. It 
would develop a maximum brake horse-power of 85 without 
losing more than 17% of its best efficiency, and could de- 
velop as little as 30 horse-power without falling below an 
efficiency now attained only at the maximum capacity of 
the actual engine adopted as a basis for comparison. It 
could develop as little as 20 horse-power without falling 
below the fuel-rate which it demanded at its maximum 
capacity, and which is the same as that of the best existing 
engines under rated load. 

The curve N B is of the same general form as that which 
has always been revealed by the standard steam-engine. 
It expresses characteristics which are essential to the broad 
success of any prime-mover. Their incorporation into the 
explosive gas-engine would constitute a long stride forward 
toward that mechanical parallelism with the steam-engine 
which alone can mark the completion of its growth. 

Before leaving the set of diagrams which have formed 
the text of the foregoing discussion, attention should be 



304 THE THERMODYNAMICS OF HEAT-ENGINES 

given to the entropy-diagrams, Figs. 51, 53, 55, and 57. 
There lies in them much food for instructive thought. 
The only point to which particular notice is to be called at 
present is the persistent concavity (toward the right) of the 
expansion-lines, in spite of the greatest variety of form, 
from sharp peak to flatly rounded top, on the part of the 
pressure-volume curve. This, and the approach of the 
later stages of expansion to isothermal conditions, hold 
true of practically every card the writer has ever analyzed. 
They leave no room for doubt as to the entire completion 
of combustion almost immediately after ignition and the 
sole activity of cylinder-wall influence in the deformation 
of the expansion-line from a true adiabatic. They also 
prove beyond question that the indicator-card alone, 
untranslated into the entropy-temperature diagram, throws 
about as brilliant a light upon the real nature of the pro- 
cesses transpiring within the cylinder as would a German 
work upon thermodynamics to one who knew no German. 
The isothermal character of the expansion in its later 
stages would go to show that the gain in efficiency due to 
prolonging it by means of variable clearance might easily 
be greater than that calculated above. 



APPENDIX 



APPENDIX 

TABLE OF SPECIFIC AND LATENT HEATS OF 
VARIOUS SUBSTANCES 

SPECIFIC HEATS 
Solids 



Aluminum . 


0.21 


Iron, at 32 


O.IIO 


Silver . 


• 0.057 


Brass . 


0.092 


Iron, at 6oo° 


0.130 


Clay . 


• 0.055 


Copper, at 32 


0.090 


Tin . 


0-055 


Stone . 


. 0.20 


Copper, at 6oo° 


0.099 


Zinc . 


0.095 


Glass . 


. 0.18 


Mercury 


0.033 


Lead .. 


0.030 


Ice 


• 0.50 



Liquids 



Anhydrous ammonia 
Bisulphide of carbon 
Ether 



1.23 
0.24 

o.53 



Alcohol 
Oils . 



. 0.6 
0.4 to 0.5 



Gases 





Under 
Const. Press. 


Under 
Const. Vol. 


Ratio 


Density 


Atmospheric air 


O.237 


O.169 


I.4024 


I. OOO 


Ammonia (anhydrous) . 


O.52 


O.4I 


I.27 


0.59 


Bisulphide of carbon 


O.I6 








I.266 


Carbonic acid 


O.217 


O.167 


I.30 


I.520 


Carbonic oxide 


O.242 


O.I73 


I.40 


O.967 


Ether .... 


O.47 


045 


I.04 


2-557 


Hydrogen, at o° . 


340 


2.4I 


I.4I 


O.069 


Hydrogen, at ioo° 


341 


2.42 


I.4I 





Methane 


0.593 


O.45O 


I.3I6 


0-555 


Nitrogen 


O.244 


O.I73 


I.4I 


O.97I 


Oxygen .... 


O.217 


O.I5S 


I.40 


1. 106 


Superheated steam 


0.6 


O.46 


i-3 


O.622 


Alcohol .... 


0453 


O.398 


1. 14 


I.589 



307 



3 o8 



APPENDIX 



LATENT HEATS 
Of the Melting of Solids 





Temperature of 






the Melting Point 




Cast-iron, white .... 


60 





Cast-iron, gray 










42 





Lead 










IO 


617 


Mercury 














5 


-40 


Silver 














38 


1830 


Tin . 














2 5 


414 


Zinc 














5° 


780 


Ice . 














142 


32° 


Sea-ice 














98 


1 6° 



Of the Vaporization of Liquids 

Ammonia (anhydrous) 554—578 

Bisulphide of carbon ......... 150-180 

Carbonic acid .......... 1 10 

Ether 162 

For the above values the author is chiefly indebted to the authorities cited 

in Landolt & Bornstein. He has also drawn upon the table in Pupin's 
" Thermodynamics." 



THE VARIABILITY OF THE SPECIFIC HEATS 

In the preceding work it was invariably assumed, except in the 
case of water, that the specific heats of all substances were con- 
stant for all temperatures. In the thermodynamics of superheated 
steam and of the gas- and oil-engines the truth or falsity of this 
assumption becomes important. It would not, however, affect the 
graphical methods or the general conclusions reached in the dis- 
cussions presented, and for this reason its brief treatment has been 
reserved to this point, as a refinement which might be added by 
those so inclined. 



APPENDIX 309 

Several scientific investigators, including Regnault, have main- 
tained the constancy of the specific heats under varying pressure 
and temperature. Nevertheless, the weight of evidence is over- 
whelmingly to the opposite. But the variations appear to be 
peculiar to each gas investigated, and no general laws have been 
established. It generally appears that the specific heats increase 
with both pressure and temperature, at least for all gases of interest 
to the engineer. The maximum is reached at a pressure of from 
350 to 400 atmospheres ; with temperatures it seems to rise indefi- 
nitely. A few experimental results are given, to indicate the state 
of the science. 

The Permanent Gases. — As these become of interest chiefly in 
connection with gas-engine problems, the reader is referred to the 
able work done by Professor Burstall for the Gas-engine Research 
Committee of the Brit. Inst. M. E. Professor Burstall quotes the 
results given by Mallard and Le Chatelier for S v as : — 

Oxygen, . . 0.1488 + 0.0000763 /; 

Nitrogen, . . 0.170 -f- 0.0000872 /; 

Carbonic acid, . 0.1477 + 0.000176 t; 

Superheated steam, 0.321 + 0.000219 /; 

wherein / is temperature Fahrenheit. 

Professor Burstall adds the conclusions which he reached as to 
the variability of the specific heat of gas-engine mixtures in terms 
of the richness of mixture. The writer has taken the liberty of 
presenting them, for convenience, in the form of Fig. 59. It gives 
the values (on the vertical scale) of A and B in the formula — 

A B 

o„ = 0.2400 -\ h 



10,000 1,000,000 

in terms of the proportion of air to gas in the mixture by volume. 
(See also Batelli, in Rapports Presentes an Congrh International 
de Physique, 1900, and to the authorities referred to therein.) 

Superheated Steam. — Aside from the value of Mallard and 
Le Chatelier given above only one set of results is given for super- 



3io 



APPENDIX 



180 


- 
















170 


- 
















160 


- 
















150 


- 
















140 


- 
















130 


- 
















120 


B S 
















110 


< 
















100 


































B 


90 


















80 


- 
















70 


















60 




■ 
















5 


6 


/ 


8 


9 


10 


11 X 


V 13 



FIG. 59 
Proportion of Air to Gas, by Volume. 



APPENDIX 



311 



heated steam, that reported by Grindley {Phil. Trans., Vol. 194). 
These experiments are very carefully made ; they are quite recent ■ 
they should be significant. The investigator reports, as his general 
conclusions, that the specific heat under constant pressure is inde- 
pendent of the pressure, but varies with the temperature. His 
conclusions, and also his tabulation of his calculations from Hirn's 
experiments, are appended. 

VALUES OF THE MEAN SPECIFIC HEAT, UNDER CONSTANT 
PRESSURE, OF SUPERHEATED STEAM AT ATMOSPHERIC 
TEMPERATURE 

HlRN 



Temperatures between which the Specific 
Heat is taken 


Mean Specific Heat 


260.0 


to 


263.I 


0.3048 


263.I 


" 


27I.4 


0.6742 


271.4 


u 


279.9 


0.5362 


279.9 


a 


287.I 


0.5428 


287.1 


n 


29I.4 


0.7874 


291.4 


it 


296.6 


0.5839 


296.6 


a 


301.2 


-5977 


301.2 


" 


306.5 


0.9258 


306.5 


" 


3I2.0 


0.8028 


312.0 


" 


3I4-I 


0.9577 


3I4-I 




316.O 


0.9309 



Grindley 



Temperatures between which the Specific 
Heat is taken 


Mean Specific Heat 


230.7 to 246.5 
246.5 " 260.8 

260.8 " 269.7 
269.7 " 2 95-° 
295.0 " 31 1.5 


0.43I7 
0.4778 

0.5152 
0.5646 
0. 6482 



312 APPENDIX 

Because the above results do not seem to be altogether con- 
sistent, the writer has taken the liberty of recalculating the values 
which seem to him to be pointed to by the data which Grindley 
records as the result of his investigation, and without reducing 
them to atmospheric pressure. These values are displayed in 
Fig. 60, which shows that portion of the entropy-temperature, or, 
to give it its proper name, the thermal, diagram covering the field 
investigated^ SS is the saturation-curve. PPPP are constant-press- 
ure lines, their level being noted where they cross -S^S 1 . QQQQ 
are the constant-heat curves upon which Grindley conducted his 
inquiry. The temperature-scale is in degrees Fahrenheit. 

Had Grindley chosen a fixed set of lower pressures to which 
to wire-draw the steam, the field would have been checkered by 
these Pand Q lines, between each pair of intersections of which 
would have been known the specific heat with great accuracy, 
averaged over a comparatively slight distance, viz., that between 
the lines. As it is, the results given are located in no regular dis- 
tribution over the field. They are most of them, too, not only 
averages over quite extended temperature-ranges, but over ranges 
which lap one over the other. Nevertheless, enough is shown 
definitely to prove conclusively : — 

(1) That the specific heat is not independent of pressure, but 
varies directly with it. 

(2) That it varies directly with the temperature. 

(3) That it varies inversely with the entropy. Lines of con- 
stant specific heat cannot be located definitely ; but it is clear 
that they would have a generally northeasterly trend. 

Whether law (3) above, taken with the hypothesis that entropy 
is a close function of molecular mass, has any connection with the 
relation between specific heat and molecular weight is a question 
for the physicists to answer. 

In view of the important role which superheated steam is begin- 
ning to play in the steam-engineering industry, the need for further 
investigation is obvious. 



APPENDIX 



313 



S IIP 




Fig. 60 1 








.a 








CTj 





Xi 


a 


hn 









w 


ni 




C 






C3 


S 






c 


V 




'V 


H 


0) 




,« 






>x 







J3 


Pi 






o c 



290 JS P* 



<U <L> qj jm 

4-1 S ctf tu 

s 2 e * 

* - rt o 

£ SI « »- 



5 »2 





JA 






H 


O 








**2 


"3 


.s° 



^ ^ a 



V 




£ 


.5 


>^ 










&H 


'* 


B 


Ph 


s 




O 
t/i 

a 
is 




2 


■y 


Hi 


P 


(33 


I) 






M rfl 


S 

u 

JA 


H 


n 


(J3 


IH 


C/J 


U 


H 



314 



APPENDIX 



A TABLE OF NATURAL LOGARITHMS 

Note. — The student is reminded that in using natural logarithms the 
logarithm of a number ten or one hundred times as great as any number 
contained in the table cannot be gotten, as in common logarithms, by adding 
one or two integers, which are the common logarithms of 10 and IOO, to the 
tabulated logarithm. Instead, to the tabulated logarithm must be added one 
or two or more times the natural logarithm of io, which is 2.302585. 



1.00 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


0.00000 


0.00 1 00 


0.00200 


0.00300 


0.00399 


0.00499 


0.00598 


0.00698 


0.00797 


0.00896 


1.01 


0.00995 


0.01094 


0.0 1 193 


0.01292 


0.01390 


0.01489 


0.01587 


0.01686 


0.01784 


0.01882 


1.02 


0.01980 


0.02078 


0.02176 


0.02274 


0.02372 


0.02469 


0.02567 


0.02664 


0.02762 


0.02859 


1.03 


0.02956 


0.03053 


0.03150 


0.03247 


0-03344 


0.03440 


0-03537 


0.03633 


0.03730 


0.03826 


1.04 


0.03922 


0.04018 


0.041 14 


0.04210 


0.04306 


0.04402 


0.04497 


0.04593 


0.04688 


0.04784 


1.05 


0.04879 


0.04974 


0.05069 


0.05164 


0.05259 


0.05354 


0.05449 


0.05543 


0.05638 


0.05733 


1.06 


0.05827 


0.05921 


0.06015 


0.061 IO 


0.06204 


0.06298 


0.06391 


0.06485 


0.06579 


0.06672 


1.07 


0.06766 


0.06859 


0.06953 


0.07046 


0.07139 


0.07232 


0.07325 


0.07418 


0.075 1 1 


0.07603 


1.08 


0.07696 


0.07789 


0.07881 


0.07974 


0.08066 


0.08158 


0.08250 


0.08342 


0.08434 


0.08526 


1.09 


0.08618 


0.08709 


0.08801 


0.08893 


0.08984 


0.09075 


0.09167 


0.09258 


0.09349 


0.09440 


1.10 


0.09531 


0.09622 


0.09713 


0.09803 


0.09894 


0.09985 


0.10075 


0.10165 


0.10256 


0.10346 


1.1 


0.09531 


0.1044 


0.1133 


0.1222 


0.1310 


0.1398 


0.1484 


0.1570 


0.1655 


O.I739 


1.2 


0.1823 


0.1906 


0.1988 


0.2070 


0.2151 


0.2231 


0.231 1 


0.2390 


0.2469 


0.2546 


1.3 


0.2624 


0.2700 


0.2776 


0.2852 


0.2927 


0.3001 


0-3075 


0.3148 


0.3221 


0.3293 


1.4 


o.33 6 S 


0-3436 


0-3507 


o-3577 


0.3646 


0.3716 


0.3784 


0.3853 


0.3920 


0.3988 


1.5 


0.4055 


0.41 21 


0.4187 


0.4253 


0.4318 


0.4382 


o.4447 


0.45 ' ! 


0-4574 


0.4637 


1.6 


0.4700 


0.4762 


0.4824 


0.4886 


0.4947 


0.5008 


0.5068 


O.5128 


0.5188 


0.5247 


1.7 


0.5306 


0.5365 


0.5423 


0.5481 


0-5539 


o.5596 


0.5653 


O.57IO 


0.5766 


0.5822 


1.8 


0.5878 


o.5933 


0.5988 


0.6043 


0.6098 


0.6152 


0.6206 


O.6259 


0.6313 


0.6366 


1.9 


0.6418 


0.6471 


0.6523 


o.6575 


0.6627 


0.6678 


0.6725 


O.6780 


0.6831 


0.6881 


2.0 


0.6931 


0.6981 


0.7031 


0.7080 


0.7129 


0.7178 


0.7227 


0.7275 


0.7324 


0.7372 


2.1 


0.7419 


0.7467 


o.75H 


0.7561 


0.7608 


0.7655 


0.7701 


O.7747 


0-7793 


0.7839 


2.2 


0.7884 


0.7930 


0-7975 


0.8020 


0.8065 


0.8109 


0.8154 


O.8198 


0.8242 


0.8286 


2.3 


0.8329 


0.8372 


0.8416 


0.8459 


0.8502 


0.8544 


0.8587 


O.8629 


0.8671 


0.8713 


2.4 


o-8755 


0.8796 


0.8838 


0.8879 


0.8920 


0.8961 


0.9002 


O.9O42 


0.9083 


0.9123 


2.5 


0.9163 


0.9203 


0.9243 


0.92S2 


0.9322 


0.9361 


0.9400 


0.9439 


0.9478 


o.95 1 7 


2.6 


o-9555 


0.9594 


0.9632 


0.9670 


0.9708 


0.9746 


0.9783 


O.982I 


0.9858 


0.9895 


2.7 


0-9933 


0.9969 


1.0006 


1.0043 


1.0080 


1.0116 


1.0152 


i. 01 88 


1.0225 


1 .0260 


2.8 


1.0296 


1.0332 


1.0367 


1.0403 


1.0438 


1 -0473 


1.0508 


1-0543 


1.0578 


1.0613 


2.9 


1.0647 


1.0682 


1.0716 


1.0750 


1.0784 


1. 081 8 


1.0852 


1.0886 


1.0919 


I-0953 


3 


1.0986 


1.1019 


1. 1053 


1. 1086 


1.1119 


1.1151 


1.1184 


1.1217 


1 . 1 249 


1. 1282 


3.1 


1.1314 


1. 1346 


1-1378 


1.1410 


1. 1442 


1. 1474 


1. 1506 


I-I537 


1-1569 


1. 1600 


3.2 


1. 1632 


1. 1663 


1. 1 694 


1. 1725 


1. 1756 


1. 1787 


1.1817 


1. 1848 


1. 1878 


1. 1909 


3.3 


I-I939 


1. 1969 


1.2000 


1.2030 


1.2060 


1.2090 


1. 21 19 


1. 2149 


1. 2179 


1.2208 


3.4 


1.2238 


1.2267 


1.2296 


1.2326 


I - 2 355 


r.2384 


1.2413 


1.2442 


1.2470 


1.2499 


3.5 


1.2528 


»- 2 55 6 


1.2585 


1. 2613 


1. 2641 


1.2669 


1.2698 


1.2726 


1-2754 


1.2782 


3.6 


1.2809 


1.2837 


1.2865 


1.2892 


1.2920 


1.2947 


1.2975 


1.3002 


1.3029 


1-3056 



APPENDIX 








1 


2 


3 


4 


5 


6 


7 


8 


9 


3.7 


i-3o83 


1.3110 


I.3I37 


1.3164 


1.3191 


1. 32 1 8 


1-3244 


1.3271 


1-3297 


L3324 


38 


1-335° 


I-337 6 


1-3403 


1-3429 


J -3455 


1. 348i 


J -3507 


1-3533 


1-3558 


1.3584 


3.9 


1.3610 


I-3635 


..3661 


1.3686 


1.3712 


!-3737 


1.3762 


1.3788 


i-38i3 


1.3838 


4.0 


1.3863 


1.3888 


I.39I3 


1.3938 


1.3962 


I-3987 


1.4012 


1.4036 


1. 406 1 


1.4085 


4.1 


1.4110 


I-4I34 


I-4I59 


1.4183 


1.4207 


1. 423 1 


I-4255 


1.4279 


14303 


14327 


4.2 


I-435 1 


M375 


1.4398 


1.4422 


1.4446 


1.4469 


1-4493 


1.4516 


1.4540 


14563 


4.3 


1.4586 


1.4609 


14633 


1.4656 


1.4679 


1.4702 


i-47 2 5 


1.4748 


1.4770 


1-4793 


4.4 


1.4816 


1.4839 


1.4861 


1.4884 


1.4907 


1.4929 


I-495 1 


r.4974 


1.4996 


1-5019 


4.5 


1. 5041 


1.5063 


1.5085 


1.5107 


1.5129 


I-5I5I 


i.5 J 73 


'■5>95 


1.5217 


I-5239 


4.6 


1.5261 


1.5282 


I.5304 


1.5326 


1-5347 


I-5369 


I-5390 


I-54I2 


1-5433 


1-5454 


4.7 


1-5476 


1-5497 


I-55I8 


'•5539 


1.5560 


i-558i 


1.5602 


1-5623 


1.5644 


1.5665 


4.8 


1.5686 


1-5707 


1.5728 


1.5748 


I-5769 


1 -5 79o 


1.5810 


1-5831 


1.585 1 


1.5872 


4.9 


1.5892 


!-59i3 


t-5933 


L5953 


1-5974 


1.5994 


1.6014 


1.6034 


1.6054 


1.6074 


5.0 


1 .6094 


1.6114 


1 .61 34 


1-6154 


1.6174 


1.6194 


1.6214 


1.6233 


1.6253 


1.6273 


5.1 


1.6292 


1.6312 


1.6332 


1.6351 


1 .637 1 


1.6390 


1 .6409 


1.6429 


1.6448 


1.6467 


5.2 


1.6487 


1.6506 


[.6525 


1.6544 


1.6563 


1.6582 


1. 6601 


1.6620 


1.6639 


1.6658 


5.3 


1.6677 


1.6696 


1-6715 


1-6734 


1.6752 


1.6771 


[.6790 


1.6808 


1.6827 


1.6845 


5.4 


1.6864 


1.6882 


1 .6901 


1.6919 


1.6938 


1.6956 


1.6974 


1.6993 


1. 701 1 


1.7029 


5.5 


1.7047 


1.7066 


[.7084 


1. 7102 


1. 7120 


1-7138 


1.7156 


1.7174 


1.7192 


1. 7210 


5.6 


1.7228 


1.7246 


1.7263 


1.7281 


1.7299 


!-73i7 


*-7334 


I-735 2 


1-737° 


L7387 


5.7 


I-7405 


1.7422 


I-7440 


1-7457 


!-7475 


1.7492 


1 -75°9 


I-7527 


1-7544 


1.7561 


5.8 


1-7579 


1-7596 


1-7613 


1.7630 


1.7647 


1.7664 


1. 7681 


1.7699 


1.7716 


'•7733 


5.9 


I -775° 


1.7766 


t.7783 


1.7800 


i-78i7 


1-7834 


1.7851 


1.7867 


1.7884 


1. 7901 


6.0 


1.7918 


1-7934 


I-795I 


1.7967 


1.7984 


1. 8001 


1. 8017 


1.8034 


1.8050 


1.8066 


61 


1.8083 


1.8099 


r.8116 


1.8132 


1. 8148 


i.8i6 S 


1.8181 


1.8197 


1.8213 


1.8229 


6.2 


1.8245 


1.8262 


1.8278 


1.8294 


1.8310 


1.8326 


1.8342 


1.8358 


1.8374 


1.8390 


6.3 


1.8405 


1. 842 1 


[-8437 


1.8453 


1.8469 


1.8485 


1.8500 


1.8516 


1.8532 


1.8547 


6.4 


1.8563 


I-8579 


[.8594 


1.8610 


1.8625 


1. 8641 


1.8656 


1.8672 


1.8687 


1.8703 


6.5 


1.8718 


1-8733 


1.8749 


1.8764 


1.8779 


1.8795 


1.8810 


1.8825 


1.8840 


1.8856 


6.6 


1. 887 1 


1.8886 


[.8901 


1.8916 


1.8931 


1.8946 


1. 896 1 


1.8976 


1. 899 1 


1.9006 


6.7 


1. 902 1 


1.9036 


r.9051 


1.9066 


1 .908 1 


1.9095 


1.9110 


1.9125 


1. 9 140 


I.9I55 


6.8 


1.9169 


1.9184 


[.9199 


i.93i3 


1.9228 


1.9242 


L9257 


1.9272 


1.9286 


1. 930 1 


69 


i-93i5 


i-933o 


[-9344 


1-9359 


!-9373 


1-9387 


1.9402 


1. 941 6 


1-943° 


1-9445 


70 


1-9459 


1-9473 


[.9488 


1.9502 


1.9516 


1 -95 30 


1-9544 


r -9559 


1-9573 


I-9587 


7.1 


1. 9601 


1.9615 


[.9629 


1.9643 


1-9657 


1. 967 1 


1.9685 


1.9699 


i-97 I 3 


1.9727 


7.2 


1.9741 


!-9755 


[.9769 


1.9782 


1.9796 


1.9810 


1.9824 


1.9838 


1. 985 1 


1.9865 


7.3 


1.9879 


1.9892 


[.9906 


1.9920 


t-9933 


1-9947 


1.9961 


1.9974 


1.9988 


2.0001 


7.4 


2.0015 


2.0028 


2.0042 


2.0055 


2.0069 


2.0082 


2.0096 


2.0109 


2.0122 


2.0136 


7.5 


2.0149 


2.0162 


2.0176 


2.0189 


2.0202 


2.0215 


2.0229 


2.0242 


2.0255 


2.0268 


7.6 


2.0281 


2.0295 


2.0308 


2.0321 


2-0334 


2.0347 


2.0360 


2-0373 


2.0386 


2.0399 


77 


2.0412 


2.0425 


2.0438 


2.0451 


2.0464 


2.0477 


2.0490 


2.0503 


2.0516 


2.0528 


7.8 


2.0541 


2.0554 


2.0567 


2.0580 


2.0592 


2.0605 


2.0618 


2.0631 


2.0643 


2.0656 


7.9 


2.0668 


2.0681 


2.0694 


2.0707 


2.0719 


2.0732 


2.0744 


2-0757 


2.0769 


2.0782 


8.0 


2.0794 


2.0807 


2.0819 


2.0832 


2.0844 


2.0857 


2.0869 


2.0881 


2.0894 


2.0906 


8.1 


2.0919 


2.0931 


2.0943 


2.0956 


2.0968 


2.0980 


2.0992 


2.1005 


2.1017 


2.1029 


82 


2.1041 


2.1054 


2.1066 


2.1078 


2.1090 


2.1 102 


2.1114 


2.1 126 


2.1 138 


2.1150 


8.3 


2.1163 


2.1175 


2.1187 


2.1199 


2.1211 


2.1223 


2.1235 


2.1247 


2.1258 


2.1270 



3i6 



APPENDIX 








1 


2 


3 


4 


5 


6 


7 


8 


9 


8.4 
8.5 
8.6 


2.1282 
2.1401 
2.1518 


2.1294 
2.1412 
2.1529 


2.1306 
2.1424 
2.1541 


2.1318 
2.1436 
2.1552 


2.1330 
2.1448 
2.1564 


2.1342 

2.H59 
2.1576 


2.1353 
2.1471 
2.1587 


2.1365 
2.1483 
2.1599 


2.1377 
2.1494 
2.1610 


2.1389 
2.1506 
2.1622 


8.7 
8.8 
8.9 


2.1633 
2.1748 
2.1861 


2.1645 

2.1759 
2.1872 


2.1656 
2.1770 
2.1883 


2.1668 
2.1782 
2.1894 


2.1679 

2.1793 
2.1905 


2.1691 
2. 1 804 
2.1917 


2.1702 
2.1815 
2.1928 


2.1713 

2.1827 
2.1939 


2.1725 
2.1838 
2.1950 


2.1736 
2.1849 
2.1961 


9.0 


2.1972 


2.1983 


2.1994 


2.2006 


2.2017 


2.2028 


2.2039 


2.2050 


2.2061 


2.2072 


9.1 
9.2 
9.3 


2.2083 
2.2192 
2.2300 


2.2094 
2.2203 
2.231 1 


2.2105 
2.2214 
2.2222 


2.21 16 

2.2225 
2.2332 


2.2127 
2.2235 
2.2343 


2.2138 
2.2246 
2.2354 


2.2148 
2.2257 
2.2364 


2.2159 
2.2268 

2-2375 


2.2170 
2.2279 
2.2386 


2.2181 
2.2289 
2.2396 


9.4 
9.5 
9.6 


2.2407 
2.2513 
2.2618 


2.2418 
2.2523 
2.2628 


2.2428 
2.2534 
2.2638 


2.2439 
2.2544 
2.2649 


2.2450 

2-2555 
2.2659 


2.2460 
2.2565 
2.2670 


2.2471 
2.2576 
2.2680 


2.2481 
2.2586 
2.2690 


2.2492 

2.2597 
2.2701 


2.2502 
2.2607 
2.271 1 


9.7 
9.8 
9.9 


2.2721 
2.2824 
2.2925 


2.2732 
2.2834 
2.2935 


2.2742 
2.2844 
2.2946 


2.2752 
2.2854 
2.2956 


2.2762 
2.2865 
2.2966 


2.2773 
2.2875 
2.2976 


2.2783 
2.2885 
2.2986 


2.2793 
2.2895 
2.2996 


2.2803 
2.2905 
2.3006 


2.2814 
2.2915 
2.3016 


10.0 


2.3026 


2.3036 


2.3046 


2.3056 


2.3066 


2.3076 


2.3086 


2.3096 


2.3105 


2.3115 



THE STEAM-TABLE 



A TABLE OF THE THERMAL AND PHYSICAL 
PROPERTIES OF SATURATED STEAM- 
VAPOR AND OF THE SPECIFIC 
HEAT OF WATER 



COMPILED FROM VARIOUS SOURCES BY 

SIDNEY A. REEVE 

PROFESSOR OF STEAM-ENGINEERING AT THE WORCESTER 
POLYTECHNIC INSTITUTE 



Wefo f£<rrft 
THE MACMILLAN COMPANY 

LONDON : MACMILLAN & CO., Ltd. 
1903 

All rights reserved 



Copyright, 1903, 
By THE MACMILLAN COMPANY. 



Set up and electrotyped January, 1903. 



Nortoocti 3l«Bg 

J. S. dishing & Co — Berwick & Smith 

Norwood Mass. U.S.A. 



THE STEAM-TABLE 

NOTES EXPLANATORY OF THE STEAM- 
TABLE 

A brief investigation of the present knowledge of the 
thermal properties of water and steam leaves much to be 
desired. The various authorities are far from agreement. 
In consulting them, either all except one must be discarded 
as worthless, or else a highly unscientific compromise or 
average must be effected in order to recognize them all. 
The following table represents a combination of both 
courses. Nothing more can be claimed for it than a con- 
scientious effort after probability of truth. 

The British thermal unit in which it is expressed is the 
specific heat of water at 59 Fahrenheit or 15 Centigrade. 
The mechanical equivalent of heat is taken as 778. The 
weight of one cubic inch of mercury is taken as 0.4912 
pound. 

In making use of the steam-table it is first to be noted 
that the high temperatures and pressures are at the top 
and the lower ones at the bottom. The index figures from 
which the table is to be entered are in pressure and tem- 
perature both, and are printed in heavy type in the middle 
of the page. Either index may be used, according to con- 
venience. From a given index the desired value will be 
found by passing horizontally to right or left. Immedi- 
ately above the value sought is printed, in small type, the 
difference needed to carry the value to that for the next 



integral index above the one chosen and of the same sort. 
For instance, if it be elected to make use of a pressure- 
index, and 21 pounds per square inch be the integral press- 
ure involved, opposite 21 pounds will be found the values 
for that pressure. Above them are the differences required 
to carry those values to the correct ones for 22 pounds per 
square inch, irrespective of the fact that the values for 
22 pounds lie several lines above. By the aid of this dif- 
ference any fractional pressures between 21 and 22 pounds 
may easily be interpolated. 

The arrangement of the indexes in pressure and tem- 
perature side by side will be found to reduce the need for 
and the work of interpolation to a minimum. 

All pressures near or below atmospheric are expressed 
in inches of mercury-column, thus avoiding the necessity 
for translation of observations into pounds per square inch ; 
at the same time the gaps which involve the work and the 
error of interpolation are reduced by half. 

The column of logarithms at the right will be found con- 
venient in handling temperature-ratios, both for entropy 
calculations and for ordinary purposes. While these are 
themselves logarithms of ratios, it is plain that the dif- 
ference between any two is itself the logarithm of the ratio 
of the corresponding temperatures. This column, in fact, 
reduces all temperature-calculations to the basis of a scale 
in which the absolute temperature of melting ice is unity, 
which is the only natural temperature-scale. 



QUANTITIES OF HEAT-ENEBGY 


DIMENSIONS 


Natural 
Loga- 




Sensible 
Heat 
of the 
Liquid 
above 
32° F, 


Disgre- 
gation 
Energy 


Heat- 
Equiva- 
lent of 
External 
Work 


Latent 
Heat of 
Vapori- 
zation 


Total 
Heat, 
from 
Water 
at 32° 


Of the Body 


Of the Heat-Energy 


rithms of 
the Ratio of 


Specific 
Heat of 


Volume 

Ou, Ft. 

per 
Pound 


Pressure 

Lbs, per 

Sq. In. 

abs. 


Temperatures 


Entropies 


1 Absolute 
| Tempera- 
ture to that 
of Melting 
Ice 


Water 


Fahr, 


Abs. 


Of the 
Water 


Of Vapori- 
zation 


Of Satur- 
ated Steam 


S 


1 


D 


X 


t 


H 


V 


P 


t 


T 


N w 


N v 


N s 


X T 



2 


1.1 


7 


! 


7 




11 


4.0 




1.0 


11 


!7 


6 




I.O562 


420.6 


711.9 


85.I8 


797.1 


1217.7 


1. 167 


4OO.9 


445 


905.8 


0.6184 


0.8800 


I.4984 




O 


2 


2 


O 


2 


1 


3 




2 


2 


3 


4 


2 




I. O56I 


420.4 


7I2.I 


85.I8 


797-3 


1217.6 


1. 170 


400 


444.8 


905.6 


0.6181 


O.8804 


1.4985 




I 


3 


2 


O 


2 


1 


3 




3 


3 


2 


4 


2 




I.O56O 


420.1 


712.3 


85.I8 


797-5 


1217.5 


I-I73 


399 


444-5 


905-3 


0.6179 


O.8808 


1.4987 




O 


2 


I 


I 


I 


O 


3 




2 


2 


4 


5 


I 




I.O56O 


419.9 


712.4 


85-I9 


797.6 


1217.5 


1.176 


398 


444-3 


905.1 


0.6175 


O.8813 


1.4988 




I 


3 


2 


O 


2 


1 


3 




3 


3 


3 


4 


2 




I-0559 


419.6 


712.6 


85.19 


797.8 


1217.4 


1. 179 


397 


444.0 


904.8 


0.6172 


O.8817 


1.4990 




3 


1.0 


7 


I 


7 


3 


12 


4-i 




1.0 


12 


18 


6 




I-0559 


419.6 


712.6 


85.19 


797.8 


1217-4 


I.179 


396.8 


444 


904.8 


0.6172 


O.8818 


I.4990 




O 


2 


2 


O 


2 


I 


3 




2 


2 


2 


4 


I 




1-0559 


419.4 


712.8 


85.19 


798.0 


I2I7-3 


1.182 


396 


443-8 


904.6 


0.6170 


0.8821 


i. 499 1 




I 


3 


I 


O 


2 


1 


3 




3 


3 


3 


5 


2 




I.0558 


4I9-I 


712.9 


85.19 


798.2 


1217.2 


1.185 


395 


443-5 


904-3 


0.6167 


O.8826 


1-4993 




I.0558 


418.8 


7I3-I 


85.20 


798.3 


1217.1 


1. 188 


394 


443-3 


904.1 


3 
0.6164 


O.8830 


1.4994 




I 


2 


2 


O 


2 


1 


3 




3 


3 


3 


4 


2 




1-0557 


418.6 


7I3-3 


85.20 


798.5 


1217.0 


1. 191 


393 


443-o 


903.8 


0.6161 


O.8834 


1.4996 




2 




7 


I 


7 


4 


12 


4.0 




1.0 


12 


*7 


6 




j-oss? 


418.5 


7I3-3 


85.20 


798.5 


1 21 7.0 


1. 191 


392.8 


443 


903.8 


0.6160 


O.8835 


I.4996 







3 


I 





2 


1 


3 




2 


2 


3 


5 


I 




'•OSS? 


418.3 


7 J 3-4 


85.20 


798.7 


1216.9 


1.194 


392 


442.8 


903.6 


0.6158 


O.8839 


1.4997 




I 


3 


2 





I 





3 




2 


2 


3 


4 


2 




1-0556 


418.0 


7 r 3-6 


85.20 


798.8 


1216.9 


1.197 


391 


442.6 


903-4 


0.6155 


O.8843 


1.4999 







2 


2 


I 


2 


1 


3 




3 


3 


3 


5 


1 




1-0556 


417.8 


713-8 


85.21 


799.0 


1216.8 


1.200 


390 


442.3 


903.1 


0.6152 


O.8848 


1.5000 




I 


3 


2 





2 


1 


3 




2 


2 


3 


4 


2 




'•t^ 


417-5 


714.0 


85.21 


799.2 


1216.7 


1.203 


389 


442.1 


902.9 


0.6149 


O.8852 


1.5002 




3 




7 




7 


3 




4.0 








18 


6 




I-0554 


417-5 


714.0 


85.21 


799.2 


1216.7 


1.203 


388.8 


442 


902.8 


0.6149 


O.8853 


I.5002 




I-0554 


417.2 


714.1 


85.21 


799-3 


1216.6 


3 
1.206 


388 


441.8 


3 
902.6 


0.6147 


O.S856 


I.5OO3 




I 


2 


2 





2 


1 


3 




3 


3 


3 


5 


1 




I-0553 


417.0 


7 I 4-3 


85.21 


799-5 


1216.5 


1.209 


387 


441.5 


902.3 


0.6144 


O.S861 


I.5004 




»-°553 


1 416.7 


7H-5 


85.22 


799-7 


1216.4 


3 

1. 212 


386 


441-3 


902.1 


4 
0.6140 


O.S865 


I.5OO5 





0.60871 
0.60849 

33 
O.60816 

O.60794 

33 

O.60761 

O.60761 

O.60739 

33 

O.60706 

O.60684 

34 

O.6065O 

O.60650 

O.60628 

O.60606 

33 

O.60573 

O.6055I 

O.6054O 
34 

O.60517 
33 

O.60484 

O.60462 



- o 


MOO 


NO 


ro rO 




o O 


* r-» 


ro r}- 


o N 


K "1 


<N O) 


U (J 




« in 


" ■* 




rr 10O 


N O 


M^o 


SO 


SO 


in 


in 


>n 


in 


tJ- 


it 


O 


o 


O 


(5 


() 


o 


CI 


n 


n 


\o 


SO 


so 


SO 


SO 


so 


so 


SO 


SO 


o 


o 


O 


o 


o 


o 


o 





o 


3VO 


h r^ 


N OS 


H R 


N f» 


a N 


h ro 


M •«*■ 


m in 


On 


ON 


OS 


o 


O 


O 


n 


C) 


OS 


Os 


Os 


O 


<) 


O 


<) 


o 


o 


"<*■ 


sj- 


rj- 


in 


in 


in 


in 


in 


in 


>.in 


mON^rO inCO 


■* N 


x> ro tj-sO 


o« 


-+ in 


- ro 


ro 


^ 


"* 


in 


H U-) 


in 


O 


so 


00 


00 


CO 


00 


00 


00 


00 


00 


00 


00 


00 


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t^t^t~»t^r^r^t^t^t^t^t^r^NONONONONO inio 
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40 

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ONCAC^CT\C^O\CnCjNC>iChONCAG\ 



42 



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" 



A Laboratory Flanual 

OF 

Physics and Applied Electricity. 

ARRANGED AND EDITED BY 

EDWARD L. NICHOLS, 

Professor of Physics in Cornell University . 
IN TWO VOLUMES. 

Vol L JUNIOR COURSE IN GENERAL PHYSICS. 

BY 

ERNEST MERRITT and FREDERICK J. ROGERS. 

Cloth. $3.00. 

Vol. H. SENIOR COURSES AND OUTLINE OF 
ADVANCED WORK. 



GEORGE S. MOLER, FREDERICK BEDELL, HOMER J. HOTCHKISS, 
CHARLES P. MATTHEWS, and THE EDITOR. 

Cloth, pp. 444. $3.25. 



The first volume, intended for beginners, affords explicit directions adapted to a modern 
laboratory, together with demonstrations and elementary statements of principles. It is 
assumed that the student possesses some knowledge of analytical geometry and of the cal- 
culus. In the second volume more is left to the individual effort and to the maturer intel- 
ligence of the practicant. 

A large proportion of the students for whom primarily this Manual is intended, are pre- 
paring to become engineers, and especial attention has been devoted to the needs of that 
class of readers. In Vol. II., especially, a considerable amount of work in applied elec- 
tricity, in photometry, and in heat has been introduced. 

COMMENTS. 

"The work as a whole cannot be too highly commended. Its brief outlines of the 
various experiments are very satisfactory, its descriptions of apparatus are excellent; its 
numerous suggestions are calculated to develop the thinking and reasoning powers of the 
student. The diagrams are carefully prepared, and its frequent citations of original 
sources of information are of the greatest value." — Street Railway Journal. 

" The work is clearly and concisely written, the fact that it is edited by Professor Nichols 
being a sufficient guarantee of merit." — Electrical Engineering . 

"It will be a great aid to students. The notes of experiments and problems reveal 
much original work, and the book will be sure to commend itself to instructors." 

— San Francisco Chronicle. 



THE MACMILLAN COMPANY 

66 FIFTH AVENUE, NEW YORK 
CHICAGO BOSTON SAN FRANCISCO ATLANTA 



THE ELEMENTS OF PHYSICS. 



EDWARD L. NICHOLS, B.S., Ph.D., 

Professor of Physics in Cornell University , 



WILLIAM S. FRANKLIN, M.S., 

Professor of Physics and Electrical Engineering at the Iowa Agricultural College, Ames, la. 
WITH NUMEROUS ILLUSTRATIONS. 

Part I. In Three Volumes: 

Vol. I. Mechanics and Heat . . Price $1.50 net. 

II. Electricity and Magnetism . " $1.90 net. 

III. Sound and Light ..." $1.50 net. 



It has been written with a view to providing a text-book which shall correspond with 
the increasing strength of the mathematical teaching in our university classes. In most of 
the existing text-books it appears to have been assumed that the student possesses so 
scanty a mathematical knowledge that he cannot understand the natural language of 
physics, i.e., the language of the calculus. Some authors, on the other hand, have assumed 
a degree of mathematical training such that their work is unreadable for nearly all under- 
graduates. 

The present writers having had occasion to teach large classes, the members of which 
were acquainted with the elementary principles of the calculus, have sorely felt the need of 
a text-book adapted to their students. The present work is an attempt on their part to 
supply this want. It is believed that in very many institutions a similar condition of affairs 
exists, and that there is a demand for a work of a grade intermediate between that of the 
existing elementary texts and the advanced manuals of physics. 

No attempt has been made in this work to produce a complete manual or compendium 
of experimental physics. The book is planned to be used in connection with illustrated 
lectures, in the course of which the phenomena are demonstrated and described. The 
authors have accordingly confined themselves to a statement of principles, leaving the 
lecturer to bring to notice the phenomena based upon them. In stating these principles, 
free use has been made of the calculus, but no demand has been made upon the student 
beyond that supplied by the ordinary elementary college courses on this subject. 

Certain parts of physics contain real and unavoidable difficulties. These have not been 
slurred over, nor have those portions of the subject which contain them been omitted. It 
has been thought more serviceable to the student and to the teacher who may have occa- 
sion to use the book to face such difficulties frankly, reducing the statements involving 
them to the simplest form which is compatible with accuracy. 

In a word, the Elements of Physics is a book which has been written for use in such 
institutions as give their undergraduates a reasonably good mathematical training. It is 
intended for teachers who desire to treat their subject as an exact science, and who are 
prepared to supplement the brief subject-matter of the text by demonstration, illustration, 
and discussion drawn from the fund of their own knowledge. 



THE MACMILLAN COMPANY 

66 FIFTH AVENUE, NEW YORK 
CHICAGO BOSTON SAN FRANCISCO ATLANTA 



a 1903 



